This is what Aristotle meant in Poetics regarding katharsis, the purgation of pity and fear:
Tragedy, then, is an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, the several kinds being found in separate parts of the play; in the form of action, not of narrative; through pity and fear effecting the proper purgation of these emotions.
We certainly feel both pity and fear for Oedipus. We pity him because he is a good man, a good king, and a good husband and father. He is so good, in fact, that he suffers from hubris (excessive pride). He has become so good at solving the Sphinx's riddles that he cannot solve his most personal problems.
We pity Oedipus' childhood most of all. As a baby, his ankles were pinioned and the tendons tied together; he was left for dead by his own parents. Certainly, the shepherds and the king and queen of Corninth pity him; they save him from death and make him a prince.
We pity Oedipus's search for truth because it leads to suffering. Oedipus voraciously plays the role of prosecutor, defense, judge, and jury in his own trial to find his father's murderer. We pity that, like Oedipus, our search to uncover our identities will lead to tragedy.
We also fear that what happens to Oedipus can happen to us: we all have secrets in our families; we are all born with handicaps beyond our control; we all suffer, even when we do all our power to make good decisions. We all fear that our mythical search for our heavenly father is all in vain.
We fear that, as males, our fathers' expectations are impossible for us to live up to, to the point that we take for granted the females (mothers and wives) in our lives. We certainly fear that we will make the same mistakes as parents that our parents made with us. And, heaven forbid, we fear that we will marry someone just like our mother.