The handkerchief is what allows Iago to accuse Desdemona of infidelity, and it is offered to Othello as evidence of her adultery. Iago knows that Othello is a jealous man, and that the handkerchief, a physical object belonging to his wife, will give an intimate and personal urgency to the accusation of infidelity, even thought there is no truth to it. Iago contrives the situation and obtains the handkerchief, and seals Desdemona's fate when he tells the jealous king that is wife has been unfaithful, and offers the handkerchief as some form of proof.
The handkerchief may be seen as a symbol of purity, much like the wedding veil is a symbol of virginity. Brides wear the veil to the wedding to symbolize the hymen, and the veil is lifted when the groom kisses the bride, signifying the consummation of marriage. Desdemona's handkerchief is a symbol of her purity and innocence, and her name and image are sullied by its displacement.