Note: One thing we need to keep in mind about the so-called Oedipus trilogy is that Oedipus the King, Antigone, and Oedipus at Colonus were all staged in different years. Antigone actually appeared first, in about 442/441 BCE; Oedipus the King came next, most likely between 430 and 425 BCE; while Oedipus at Colonus was actually put on in 401 BCE by Sophocles' son, a few years after his father's death. Thus, unlike Aeschylus' Oresteia (all three plays were first staged in 458 BCE), Sophocles' so-called trilogy is an artificial one.
With that said, many literature teachers might tell us that Sophocles' primary interest in portraying Oedipus' downfall was to show that human beings are trapped in the grip of fate and that no escape is possible.
Other teachers will have us believe that Oedipus possessed some "tragic flaw" that caused his downfall. They would argue that Oedipus was too proud or that he was too hot-headed. After all, as everyone knows (wink, wink), all Greek literature can be explained if only we can identify the passages in which the author shows that the main character exhibits extreme pride or excessive anger.
Personally, I prefer to see Oedipus' downfall as a result of his insatiable desire to discover the precise truth about who killed Laius. Throughout the play, Oedipus and the audience hear many different versions of Laius' death. Oedipus hears these different versions and will not give up until he has discovered the true and accurate version. He will not stop until he ascertains without a doubt that the "many men" who were said to have killed Laius, was in fact one man: Oedipus himself.
Along those same lines, I prefer to see Oedipus' downfall as a result of his desire for the same thing that many of us want to know: who am I? Oedipus wants to know "who he is." Unfortunately, for Oedipus, the answer to that question is surely just about the most horrific thing that any man could discover. Thus, as Oedipus blinds himself, he cries out:
"You will no longer see
all those atrocious things I suffered,
the dreadful things I did! No. You have seen
those you never should have looked upon,
and those I wished to know you did not see.
So now and for all future time be dark!" (Ian Johnston translation)