Othello kills Desdemona because, deceived by Iago, he is convinced that she betrayed him with Cassio, and because he sees death as "a sacrifice" that also keeps her from deceiving other men: "But yes, she must die, or she will betray other men."
The conviction is built in stages. In Act III, Scene 3, Iago repeats Othello's words and lets silence work, and Othello begins to demand "ocular proof"; Iago answers with a dream that he attributes to Cassio and with a handkerchief that he says he saw with him. In Act IV, Scene 1, Othello, hidden, sees Cassio laugh while speaking of Bianca and thinks he is speaking of Desdemona, recognizes the handkerchief and, when Iago proposes strangling her, approves: "Good, perfect. It is a justice that pleases me."
In Act V, Scene 2, before suffocating her, he accuses her of having given the handkerchief to Cassio. Desdemona denies it and asks that Cassio be called, but Othello says that he has already confessed and that "honest Iago has carried out orders". She asks to be spared this night, then half an hour to pray, and he answers: "It is too late."
When the truth comes out, Othello does not claim hatred: he tells those present that he did not act "out of hatred", "but only for honor". Then he kills himself.