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.