Apply Socrates’s specific understanding of “wisdom” to analyze and explain Oedipus’s hubris and eventual downfall.

Socrates (c. 470–399 BCE) left no written record of his philosophy or his teachings. Everything we know about Socrates and his understanding of the qualities of wisdom come to us secondhand, primarily from the writings of his follower Plato (c. 428–347 BCE) and his student Xenophon (c. 431–354 BCE).
In Plato’s Apologia Socrates (Apology of Socrates), which recounts Socrates's defense of himself at his trial in 399 BC, Plato quotes Socrates as saying that “an unexamined life is not worth living.” In other words, it's vitally important to "know thyself," and it's equally important for a person to know what they don't know.
To "know thyself" and to discern the limits of one's knowledge, Socrates employed a question-and-answer method of elenchus (examination) attributed to him and called the "Socratic method." Socrates believed that self-knowledge occurs only in the context of a question-and-answer dialogue with others.
By this method, Socrates believed, a person does not necessarily discern what is true but realizes what cannot be true. Secondarily, a person becomes aware of their own ignorance (what they don't know), which cultivates a sense of humility in the individual.
When Teiresias reveals that it is Oedipus himself who is Laius's murderer and that Oedipus married his mother, Oedipus refuses to believe it. His hubris, his excessive pride, simply won't allow him to accept it.

OEDIPUS. Is this a plot of Creon, or thine own?

Rather than turn inward and reflect on what Teiresias tells him, Oedipus lashes out at Teiresias and challenges him, and others, to prove to him that he is, in fact, Laius's murderer.

OEDIPUS. [to Creon] Sirrah, what mak'st thou here? Dost thou presumeTo approach my doors, thou brazen-faced rogue,My murderer and the filcher of my crown?Come, answer this, didst thou detect in meSome touch of cowardice or witlessness,That made thee undertake this enterprise?

Oedipus's dialogue with Creon is not the kind of elenchus between equals that Socrates envisions, but more like a king-subject, prosecutor-witness relationship. Oedipus questions others, but he's far too prideful and far too self-protective to question himself and to question his own beliefs; he lacks that self-knowledge which, according to Socrates, is the essence of wisdom.
By the time Oedipus realizes and accepts the truth—that he's the man who killed Laius and that he married his own mother—his entire world has crumbled around him. He's a self-blinded, broken man without a father, mother, wife, or country. Although many of the events of his life were not of his own making, those decisions that Oedipus made based on hubris and his inability to know himself (and know the depth of his own ignorance) caused his downfall.

In several dialogues, including Phaedrus, Plato has Socrates state the Delphic maxim "know thyself." In Phaedrus, Socrates says he does not have time to explore mythologies or arcane theories because he does not yet know himself. Socrates statement implies that the quest for knowledge involves first cultivating a degree of humility: he cannot begin to pretend to know everything—or anything—until he first attains wisdom about himself.
Oedipus starts out in the play full of pride or hubris, believing he knows all there is to know about himself and never once considering that he could be the cause of the plague ravaging Thebes. It does not occur to him that what he thinks he knows about himself might be false. He never imagines that when he passes judgement on the source of the plague, he is passing judgement on himself.
Because Oedipus is not humble, he has never sought wisdom or self-knowledge. When he does find out the truth that he murdered his father and married his mother, the shock he experiences is extreme. It results in him blinding himself, the point at which he begins to develop true wisdom and humility.

Socrates defines wisdom as a kind of humility, an ability to understand the limitations of one's knowledge. Someone who possesses this kind of wisdom is keenly self-aware and knows that he or she should not make assumptions beyond the scope of his or her actual knowledge. However, Oedipus's pride leads him to make all kinds of assumptions well beyond the scope of what he knows, and these assumptions lead to his downfall.
When Oedipus hears of the prophecy that he will kill his father and marry his mother, he assumes that he knows better than the oracle (and, by extension, the gods) and tries to avoid the prophecy by deciding not to return home. Later, Oedipus assumes that Teiresias is holding back information, not because he wants to protect Oedipus but because he is in league with Creon, Oedipus's brother-in-law (and uncle, though he does not know it yet). Oedipus's hubris—his terrible pride—leads him to believe that he can make assumptions accurately, and so he very much lacks the kind of wisdom Socrates describes.

Socrates's philosophy of wisdom was founded strongly upon the acknowledgement that we are all fundamentally ignorant: he defended himself at his trial with the argument that he himself was the wisest man in Athens because he was sufficiently self-aware to recognize his own ignorance. In Sophocles's play, Oedipus's fatal flaw ("hamartia") could be defined as, arguably, either his tendency to act rashly and become angry or his pride. Both of these elements can be analyzed in relation to Socrates's concept of wisdom.
One might argue that Oedipus caused the prophecy to fulfill itself when he was pushed off the road by his father; Oedipus showed a lack of wisdom and restraint in killing his father, regardless of whether it was actually ordained that he do so. We could also point to his declaration that he would banish whoever had caused the plague, as this was arguably driven by his personal fury at Tiresias for keeping this knowledge to himself. In both these instances, Socrates's conception of wisdom suggests that if Oedipus had possessed greater self-awareness and self-control, he may not have been brought low.
We can also point to the fact that Oedipus is completely unaware of his limitations, something Socrates believed to be key to wisdom. Oedipus saved Thebes from the Sphinx and has been a good ruler, which has given him a high opinion of himself. This causes him difficulties when the plague strikes because he is so self-assured and proud that he is certain he can identify the cause of the plague and dispose of it. Unfortunately, this proves to be impossible. If Oedipus had acknowledged how little he truly understood about the world and his own capabilities, he might have been more careful and more successful.