Why did Othello kill his wife, Desdemona, and himself?
One reason that Othello kills both himself and his wife is out of a duty to honor.
Why, any thing: An honorable murderer, if you will; For nought I did in hate, but all in honor. (Act 5, Scene 2)
He believes she has cheated on him and this would be humiliating for a man in his position, should it be made known. If he kills her, then she cannot cheat on him again, or any man.
Yet she must die, else she'll betray more men. Put out the light, and then put out the light: If I quench thee, thou flaming minister, I can again thy former light restore, Should I repent me; but once put out thy light, Thou cunning'st pattern of excelling nature, I know not where is that Promethean heat That can thy light relume. When I have pluck'd the rose, I cannot give it vital growth again. It must needs wither: I'll smell it on the tree. (Act 5, Scene 2)
Second, he still loves her and this passage shows us he wants to preserve that innocent memory of her. If he were to let her live, he would have to live with her as a cheater and a liar. Even as she tries to tell him before she dies that she has done no wrong, he does not hear her. He is so determined to believe she had cheated with Cassio. So, he kills her to preserve the good memories of her, as well.
Convinced by Iago that Desdemona has been unfaithful to him with Cassio, Othello, overwhelmed by his sense of betrayal, murders his innocent wife.
But he quickly comes to realize that he has been deceived, and that his own weakness (or tragic flaw), has caused him to kill the thing he loves the most in the world—and for no reason, as she had never been unfaithful.
As he is, at heart, a good and honorable man, when he sees what he has done, there is really no other recourse for him. It is not even that he will be dragged back to Venice to face the jeers and punishment that society would have in store for him. It's that he cannot live with himself, knowing what he has done.