Othello gave the handkerchief to his new bride Desdemona, which she saw as a representation of the love they shared.
Othello tells Desdemona that the red threads used to embroider the ripe strawberries on the handkerchief were dyed with virgin's blood, which represents chastity.
When Othello sees Desdemona with the handkerchief, he sees it as proof of her faithfulness. This comes to symbolize Desdemona's fidelity to Othello.