How does the play reflect the idea of destiny, and how does destiny affect Oedipus?

The play shows us the folly of trying to change our destiny. That's what Oedipus tries to do, which is why his metaphorical blindness eventually leads to his literal blindness.
It's not as if Oedipus hadn't been warned. The blind prophet Tiresias—somewhat ironically—saw more clearly what would happen if Oedipus stubbornly continued to run away from his destiny. But no; proud Oedipus was determined to flee Corinth and get to the bottom of who murdered his father, when in actual fact, he was King Laius's killer all along.
Like every mortal, Oedipus's destiny has been mapped out by the gods. To defy one's destiny, therefore, is to defy the gods. And if there's one thing we know from Greek tragedy, it's that if you defy the gods, then you'll meet a sticky end.

Oedipus Rex is a play about the impossibility of escaping one's destiny. At his birth, it is prophesied that Oedipus will murder his father and marry his mother. His parents try to prevent that from happening by having him exposed on a hillside, so that he will die as infant. However, he is saved and raised in Corinth. When he too finds out about the unsettling prophecy, he flees Corinth to avoid hurting the people he thinks are his parents.
Yet despite all these attempts to defy his destiny, Oedipus kills his father on the road to Thebes, and on his arrival in the city, he marries his mother.
Pride allows Oedipus to believe he has beaten his fate. When he finds out that the people who raised him in Corinth are not his real parents and that he has indeed killed his father and married his mother, the impact on him is so devastating that he puts his eyes out. However, this blindness leads him to humility and insight.
In the end, as the chorus states, death is what releases us from our earthly destinies:

Now as we keep our watch and wait the final day, count no man happy till he dies, free of pain at last.

Destiny is central to the plot of Oedipus Rex, and one of the themes explored in the play is how much of our destiny we can take into our hands.
When Oedipus learned that he was prophesied to kill his father and marry his mother, he fled his home and made a new life in a new city because he believed that he had the power to change his destiny. However, Oedipus later learns that the father and mother he left were not his birth parents, but rather people who had adopted him after he was abandoned. In trying to flee his destiny, he ended up fulfilling it by killing his real father and marrying his real mother.
Oedipus was abandoned as a baby because his birth parents had also heard the prophecy and thought that they, too, could change their destinies. Ultimately, neither Oedipus nor his birth parents were able to change their destinies because they were always destined to try to change the prophecy and therefore end up fulfilling it.