The ethics the modern Western world has inherited from Christianity is an ethics of obligation. In this system, there are certain moral laws, and we are obligated to obey them. A failure to obey these laws represents an unwillingness on our part. If we go against the moral law, we are guilty of breaking that law. This conception of guilt draws on an ethical system wherein morality is something that can be disobeyed or resisted.
Greek ethics are based more on the notion of virtue than obligation. The Greek conception of reality is closely tied up to the concepts of goodness and harmony. This idea is clearly expressed in Plato's theory of Forms: the real world is made up of perfect, unchanging Forms, and it is our duty to approximate this reality as best we can. Virtue, for the Greeks, is a matter of attaining our real nature and of finding our true form. Thus, moral failure is not a matter of guilty recalcitrance, but simply a matter of error, of shortcoming, or of being unable for whatever reason to attain our true nature.
Hamartia, then, represents the Greek, and not the Christian, conception of moral failure. Greek heroes are not bad people—Aristotle explicitly states that they cannot be bad people—but are simply good people who fall short in some important respect. Tragedy is less a matter of showing how bad people are punished for their crimes, and is more a matter of showing how ignorance and error can have disastrous effects. The action is tragic precisely because we are all ignorant to some degree, all flawed, and we may all suffer deeply for these errors. This is a cold, hard fact of nature, and not a matter of justice and retribution.
In these sections, Aristotle is much less of an observer and much more of a legislator. He is no longer simply stating how tragedies tend to play themselves out but is now putting forward arguments as to what makes the best tragic plot. He is explicitly asking how we can maximize the feelings of pity and fear, which he calls "tragic pleasure." That he should refer to our pity and fear as "pleasure" is further evidence that he does not mean the kind of pity and fear we might experience were the events real.
However, Aristotle does seem to treat this kind of pity and fear as the goal of a good tragedy, which would contradict the commentary on Chapter 6 (which suggested that tragedians aim at more than just emotional therapy). We can perhaps answer this conundrum by treating pity and fear as a necessary means to some other end. Surely, Aristotle does not think the value of tragedy lies simply in its emotional effect but thinks rather that it lies in what these emotional effects can in turn provoke within us. This ultimate end is naturally harder to articulate, but it has something to do with a greater sense of awareness—of our shortcomings, of our fate, and of our behavior, etc. Presumably, this added awareness helps us overcome our ignorance and other shortcomings; in short, tragedy can help us with our own hamartia.
The question Aristotle focuses on, however, is how fear and pity are most effectively aroused? He suggests that the tragic hero ought to be neither overwhelmingly good nor overwhelmingly bad, but rather intermediate, much like us. We should be able to see in the hero a better version of ourselves. Our pity and fear will be aroused by the realization that if a better person than us can be made to suffer for his or her shortcomings, then we, too, may suffer for ours.