Othello, whose history is full of strange and mystical details, honestly believes that there is a strong power in this little piece of linen.
We know that both Othello and Desdemona believe that the handkerchief has certain powers, because we are told its significance even before Othello tells its story. Emilia says, "My wayward husband hath a hundred times Wooed me to steal it" (3.3.291-2), so we know that Iago has earlier knowledge of its importance to the couple's relationship. Emilia also reveals that Othello has already given Desdemona some information concerning its properties when she says, "he conjured her she should ever keep it" (3.3.293). Emilia also lets us know that Desdemona understands it as a tangible symbol of her love for Othello, because she kisses and talks to it, and "she'll run mad When she shall lack it" (3.3.315).
The point that Othello hears that Cassio has the handkerchief is the moment he starts to believe Iago, because he believes his mother's story and understands the handkerchief as a real indicator of the status of his relationship. On these words, he gives the order for Cassio's death, and although Desdemona doubts he even has the capacity for jealousy, he now confronts her with that emotion. His mother said, "To lose't or give't away were such perdition As nothing else could match" (3.4.67-8).