For the ancient Greeks, in order to answer the motivating question you need to answer a prior question: what is the good? The idea here is that before you can figure out what a good life is you need to figure out what "good" is, then you'll want to partake in the "good" as much as possible.
As it turns out, there are two types of "good": intrinsic and instrumental. Lets begin with the latter by using an example. Money is a instrumental good because we don't seek it for itself, rather we seek it for the goods we can trade it for. We seek money because with it we can get the things we're really after like vacations, food, nights out, etc... If we were stranded on a deserted island and had a bunch of money, the money would have no value at all (except maybe as toilet paper). This shows that money has no intrinsic value; that is, it has no value itself. It only has instrumental value (i.e., it only has value in so far as it allows us to get other things that we really value).
To summarize, something has instrumental value if we value it only because it gets us things that we really want. The things we really want have intrinsic value: we don't want them in order to get some other thing. Common examples of things that have intrinsic value are love, friendship, happiness, and health. We don't value these things because they get us some other thing; we value them simply for what they are. They are intrinsically good.
Notice that some things have both intrinsic and instrumental value. For example, love and friendship have intrinsic value (we value them for themselves) but they also have instrumental value: they bring us happiness which is a purely intrinsic good. We don't pursue happiness for any other reasons. It's not valued as a means to some other end. Happiness is the reason why we pursue everything else and so we say it is a purely intrinsic good.
Incidentally, by happiness Aristotle did not mean pleasure or any other emotional states. He meant something very different from what we mean by it today, but lets set that aside for the moment and assume his position: For now, all you need to know is that by "happiness" Aristotle means the process of developing and realizing human excellences. To put it another way, happiness isn't an emotional state but a way of being. Assuming this is the case, then everything we do ought to be directed at this end if we want to live a good life.
To summarize up til now: We want to know how we ought to live a good life. In order to answer that question we need to figure out what the chief good is; that thing for which all other things are pursued. Knowing this will allow us to direct our actions. As Aristotle says:
Will not the knowledge of [the chief good] then have a great influence on life? Shall we not, like archers who have a mark to aim at, be more likely to hit upon what is right?
That chief good is happiness. So, if we want to live a good life we ought to pursue things that will lead us to happiness.
In a moment, we're going to come back to this instrumental-intrinsic distinction and Aristotle's definition of happiness but I want to discuss the question that motivates most of moral philosophy, why should I be moral? The answer to this question (for the ancient Greeks) is directly related to the first question of philosophy and its prior question regarding the nature of the good.
Depending on our conception of "the good" our answers will differ. For example, if I define the good as "whatever brings me pleasure" then there doesn't seem to be any strong reason for me to be moral and none at all where self-interest and morality diverge. The only reason I would have to be moral would be to avoid the consequences of people perceiving me as immoral. Of course, there are many cases where our personal self-interest happily lines up with what is moral and so in these cases I'd derive instrumental goods from being moral (because it serves my own ends, gives me a good reputation, etc...). We want to know if I'd acquire some good from acting morally even in situations where I don't stand to benefit in any obvious way.
Lets break this down. Plato, in a passage of The Republic called "Glaucon's Challenge," asks whether acting morality has any intrinsic value; that is, in addition to the times when acting morally serves our own interests (i.e., has instrumental value) does acting morally have any value above the instrumental goods it gets us; i.e., Is there any reason to value being moral even when there are no instrumental reasons for doing so?
Two other ways of framing this issue make it clear: If you could act immorally yet always be perceived as acting morally and never get caught (i.e., get all the instrumental benefits of acting morally without any personal costs) would you personally be missing out on any good? Or if you acted morally but were perceived as acting immorally, would you gain anything of value? This is Glaucon's challenge to Socrates: Does acting morally have any intrinsic good or is it purely an instrumental good.
In one of my favorite passages in all of philosophy, after giving his argument, Adeimantus (Glaucon's brother) says
For the things said indicate that there is no advantage in my being just, if I don't also seem to be, while the labors and penalties involved are evident. But if I'm unjust, but have provided myself with a reputation for justice, a divine life is promised. Therefore, since as the wise make plain to me, 'the seeming overpowers even the truth' and is the master of happiness, one must surely turn wholly to it. [my italics]In other words, the instrumentalist position is pretty compelling. If I want to give myself the best shot at achieving happiness, it's more important to appear to be moral than it is to actually be moral. The costs of acting morally are great, especially if one is going to be perceived as immoral while doing so, and the benefits of appearing moral (even if you aren't actually being moral) are great. What good does the "pretend" moral person miss out on that the real moral person gains? Or does he not miss out on anything?
Here's where we have to take another detour in order to answer the question. It will seem round-about but bear with me (and Aristotle).
What is the function of a cup? This may sound like a strange question but if we interpret this question the way the Greeks used the word "function", it won't sound so strange. "Function" should be understood as "the attributes that make a thing the sort of thing that it is." When I ask "what is the function of a cup?" I'm asking, what properties make a cup a cup, and not something else? I'll add that even under this definition of function, E-40's lyrics "we out here trying to function" don't make much sense to me.
We might answer that what makes a cup a cup is that it is something that holds fluids and it something that is easy to drink from. The degree to which a cup fulfills its function (i.e., has the properties that make it a cup), is the degree to which the cup is a good cup. For example, a cup with a leak in it (i.e., doesn't hold fluids "excellently") isn't as good a cup as one that does doesn't leak at all. Similarly, a cup that's really awkward to drink from isn't as good a cup as one that is really easy to drink from.
Notice also that if something doesn't hold fluids or can't be drunk from, it isn't a cup! (You are welcome for the profound philosophical insight; it's what we do!). The moral of the story here is that the more excellently something fulfills its essential functions (i.e., exhibits its defining properties), the more excellent that thing is as a such-and-such AND to the degree that something fails to exhibit excellence in its defining features, that thing is not a such-and-such.
So, why are we talking about cups? Wasn't this supposed to be about moral philosophy and Aristotle? Allow me to try to explain how this fits into the puzzle: We want to know what it would require to live a maximally good life. But, if we want to know what a maximally good life is, we need to know all the possible good things there are for humans. This way, like the archer, we can "aim" our actions at them. If it turns out that being moral is one of them and has intrinsic value (not merely instrumental) then we're not going to want to skip out on this good because that would mean excluding ourselves from the maximum good possible. We'd be leaving a piece of the good out.
Before moving forward it's important revisit Aristotle's notion of happiness. Happiness is the process of developing and realizing human excellences. To put it another way, happiness isn't an emotional state but a way of being/living. Happiness, the highest good, is the process of actualizing the qualities that make us human and not something else. To put it simply: happiness=the greatest good for humans=development and exercise of excellence in those features that make humans humans rather than something else.
In order to figure out whether moral virtue has intrinsic goodness we're going to have to figure out what the function of a human being is because we need to know what we need to be excellent at to achieve maximum human goodness/happiness. (Remember by function we mean "the defining features of a human/that which makes a human a human, and not something else).
This brings us back to the humble cup. How did we distinguish between a good cup and a less good cup or not cup? We said it was according to how much something possesses its essential features (i.e., fulfills its "function"). As with the cup, if we are to distinguish between a good human and a less good human, we're going to need to know what a human's essential properties are because maximum happiness will require maximum actualization of those essential properties.
Ah! At this point hopefully some light bulbs are turning on in regards to how everything is going to fit together. Lets see if we can figure out how the intrinsic/instrumental good distinction, the "function" of a human, happiness, morality, and the good life all fit together. First let's ask what traits make a human a human. For Aristotle this is the capacity to reason (and guide our actions according to reason) and live in large communities (i.e., we are "political animals").
The degree to which we are able to live in large communities will depend in large part on our moral virtues. If everyone just runs around acting only according to self-interest ("but I gots mah rights! don't tell me what to do!") without any consideration for others (i.e., not being morally virtuous), then that community will be dysfunctional and those people will not be able to fully develop and exercise important parts their essential human-ness.
Conversely, if everyone exercises moral virtue, the community will flourish along with the particular individuals that inhabit it and people will have a good shot at fully developing their essential features. Maximal development and exercise of the human virtues is the greatest good for humans and this is what happiness consists in. People in a dysfunctional community don't and can't maximally develop the essential features of human beings. Consequentially, they are cut off from full happiness.
Aside: It comes as no surprise to an Aristotlean that the rise of the tin-foil hat "individual-rights-trump-everything" dogma correlates strongly with the demise of community and with it human happiness. Yeah, he called that 2300 years ago.
So, where does this instrumental/intrinsic stuff fit in because it seems like our reason to be moral, on this model, is instrumental. I should be moral so I can be happy, right? I initially thought this and it perplexed me. How could Aristotle have made this obvious error?
(Another) Aside: Here's a rule of thumb for reading philosophy (and life generally). When a thinker whose works have survived centuries and especially millennia seem to contain an obvious error, odds are you have misunderstood their position. In philosophy we call this epistemic humility.
My mistake was to confuse the instrumental good for why we should be moral with the intrinsic good that comes from being moral. Why should I be moral? So I have a shot a maximally good life (instrumental reasons). How do I do that? I have to "get" all the possible "goods". If I don't develop moral virtue, there's no other way for me to acquiring the particular good that comes from being morally virtuous. This is the intrinsic good that comes from being moral.
I'll repeat this because it's a little tricky: We should exercise, develop and actualize our moral virtues so that we can get the full goodness necessary that makes up full human happiness. This is an instrumental reason to be good and it is an instrumental good that comes out of being virtuous, but the particular good that we get from being moral (development of our moral virtues) cannot be obtained any other way except by being morally virtuous (i.e., this is the intrinsic goodness that comes from being moral).
Lets return to the central questions: why should we be moral? Because there is no other way to "get" the fullest human good (i.e., happiness) without actually being moral. To repeat, we might have an instrumental reason for being virtuous (so we can be happy) but there's no way to achieve full human happiness without being morally vituous because otherwise an important intrinsic good will be missing. People who aren't virtuous are not exercising and developing maximum human excellence. They are not fully developed humans and therefore never have a shot at maximum human happiness. By not being virtuous they are cut off from a possible piece of full human excellence/happiness.
Just like a cup is a cup to the degree that it exhibits the properties that make it a cup and not something else, you are only human to the degree of excellence that you exhibit those attributes that make a human a human, and not something else. The ultimate good for a cup is to possess its essential features in the most excellent way possible. It follows that the ultimate good for a human is the same. And since moral virtue is an essential feature of human beings, if we are to achieve ultimate human good (i.e., the fullest possible happiness) we must develop moral excellence (along with other kinds); otherwise we exclude ourselves from the fullest conception of human good/happiness.
Now, go be an excellent human being!