What is the insight of the story of Oedipus?

The Greeks often dealt with the quality of hubris, especially in their tragedies. For them, it meant excessive, even overweening, pride, which more than likely resulted in defiance of the gods; in ancient Greek, hubris led to nemesis, and the root of nemesis means "given what is due." In other words, the quality of hubris leads to one's downfall and utter ruin—what other punishment could there be for a characteristic that runs counter to, even leading to a disrespect of, the gods' will?
When Oedipus visited the Delphic oracle and heard the prophecy that he would kill his father and marry his mother, he heard directly from Apollo's mouthpiece. In attempting to outwit the oracle (and his fate), Oedipus decides not to return to his home in Corinth but, rather, to head toward Thebes; this way, he could not possibly fulfill the prophecy, he thinks, because his parents live in Corinth.
He does not know that he was adopted by them, having been given up for dead by his mother, the queen of Thebes, upon his birth. He has such hubris that he thinks he can simply avoid his fate—challenging the oracle and the god, Apollo—despite the fact that he is a mere mortal. If it isn't excessive pride to believe that, with one move, Oedipus can avoid his fate and reject Apollo's prophecy, then I don't know what is. The play warns its audience to avoid the quality of hubris and to respect the gods' will.

The Greek playwrights of the classical period intended to teach their audiences with their staged dramas. Sophocles is a part of this tradition and imparts an important universal message through Oedipus Rex. The insight he offers is that unchecked pride can lead one to catastrophic results, especially if it leads one to challenge the power of the gods or of fate.
Oedipus learns at the start of the play that he can cure the plague that is afflicting Thebes by solving the murder of Laius. Because Oedipus has saved the people of Thebes once before by solving the riddle of the Sphinx, he is sure he can solve this problem as well, and he calls for the blind prophet Tiresias to question him on the matter. Tiresias hesitates to inform Oedipus of what he knows, and this entire opening sequence reflects Oedipus’s pride. He believes he is more than able to save his people, and because he is so sure he misses the signs that something is amiss. He even accuses his brother-in-law and Tiresias of conspiring against him as a result of his pride.
Later we learn the truth: Oedipus himself had killed his father, Laius, in an attempt to avoid a prophecy that he would murder his father and marry his mother. Years before, he had left his parents (who were, unknown to him, adoptive parents) and moved to Thebes. By killing Laius, and marrying Jocasta, he fulfills the prophecy he tried to avoid. Thus, the second part of the universal message becomes clear: man should not attempt to challenge fate or the power of the gods. His pride led him to cross this line, and it led to the catastrophic ending.