When Oedipus arrived to Thebes he managed to crack the Sphinx's riddle he saved the city and its people. After his help the people made him king since the previous king, Laios, was murdered.
The priest explained to Oedipus how the city was cursed with the plague after Laios's death. Oedipus promises to catch the killer and bring justice while accusing Creon, queen Jocasta's brother, of murdering the king.
Oedipus asks for Tiresias's help with finding the murderer. But after he tells Oedipus that he is the killer Oedipus stays in denial. After exchanging some insults he accuses Tiresias of betraying his home by not telling him the truth. He even says that he is working with Creon attempting to dethrone him.
After being angered on Tiresias he went back home and accused Creon of teaming up to Tiresias. He accused him of murdering the king in order to take the throne. Creon continues to deny the accusations and claims he already has the wealth so he doesn't need the extra responsibility.
Eventually Creon and Oedipus agree to have Jocasta settle the matter. But she just tells them to settle down and that Creon would never do such a thing. After they calmed down she told Creon to leave and Oedipus admits that he might be the murderer. She tells him to calm down because the Oracle said Laios's son would be the killer not him. Eventually they decide to call back the witness in order to confirm whether Oedipus was the killer
Eventually Oedipus begins telling Jocasta about his oracle. His oracle said he was destined to kill his father and marry his mother. That was the reason why he left his home and came to Thebes. He talks about how he killed a man along with some other people on his way here at the same place where the king was murdered. Jocasta soon gives a description of the king which matches the man he was talking about. He begins to realize that he was the murder.