Appearance and reality are important aspects in Othello. For Othello, seeing is believing, and proof of the truth is visual. To "prove" something is to investigate it to the point where its true nature is revealed. Othello demands of Iago "Villain, be sure thou prove my love a whore, be sure of it, give me the ocular proof" (III.3, 365–366).
What Iago gives him instead is imaginary pictures of Cassio and Desdemona to feed his jealousy. As Othello loses control of his mind, these pictures dominate his thoughts. He looks at Desdemona's whiteness and is swept up in the traditional symbolism of white for purity and black for evil. Whenever he is in doubt, that symbolism returns to haunt him, and despite his experience, he cannot help but believe it.
The significance of red is love, red strawberries like red hearts on the love token handkerchief, and like the red stains from Othello and Desdemona's first night of love on the marriage sheets. Such red on white is private and dear to the heart of Othello, and he expects it to be similarly dear to his wife. It is the belief that Desdemona gave away his handkerchief that drives him to kill her.
