He got into an argument with the stranger and killed him! He proceeded with his journey to the city of Thebes. When Oedipus arrived to the entrance of Thebes he ran into a monster who would devour anyone who answered his riddle incorrectly. Thankfully he answered correctly and the monster left Thebes and Oedipus became King to the city of Thebes!
He got married to the Queen of Thebes, Jocasta, and had 4 lovley children. Many years after a plaque struck Thebes and an oracle told Oedipus that it would'nt end until the killer of King Lalo was discovered. Oedipus discovered the man he killed on his journey was King Lalo and his real father! Which meant the women he married was his mother.
When Oedipus' real mother found out she was killed herself. To punish himself Oedipus gouged out his eyes for being blinded of the truth. Oedipus wandered the country side as a begger until he died in the santuary of Colonus.