Broadly speaking, there are two reasons behind Oedipus' actions. First of all, the literal act of blinding makes explicit the metaphorical blindness that Oedipus has displayed throughout the play, and indeed throughout his whole life. To see Oedipus in such an appalling state, blood pouring down his screaming, eyeless face, really brings home to us the terrible price that Oedipus has paid for his overweening pride.
The second reason is more practical. Oedipus doesn't want to see the looks of pity, loathing, or contempt etched on the faces of people who now know the terrible truth. It is a great source of shame for Oedipus that he killed his father and married his mother, albeit inadvertently. He literally cannot look other people in the eye now that the truth has finally been revealed.