This is a good question. In order to understand that quotation, you need some background information. The work starts off with a plague. The city of Thebes is crippled with a plague on account of some curse. Oedipus, the leader of Thebes, is hard at working looking for what that curse might be. He, as a good leader, wants to save his people. Little does he know that he himself is that curse. Without his knowing, he killed his father and married his mother. On account of this impiety, Thebes is going through this plague. So, that quotation refers to Oedipus.
As for your section question, there are many important characters in the play. So, I will have to be very selective. One of the most important characters in the play is the old blind prophet, Tiresias. He is not only wise, but he is the opposite of Oedipus in some ways. Oedipus is physically able to see, but when it comes to deeper issues, he is blind. Tiresias is physically blind, but he has supernatural sight. He is the one who enables Oedipus to see the truth.