Well, it was no ordinary handkerchief. It carries significant value. Othello's mother gave him the handkerchief. It represented faithfulness in a marriage. That is why the handkerchief is so important as Othello points out to Desdemona:
Gave that handkerchief to my mother;
She cast charms and could almost read
The thoughts of people. She told her, that, while she kept
It, it would make her sweet and soften my father
Entirely to her love; but if she lost it
Or made a gift of it, my father's eye
Should see her as hateful, and his spirits should hunt
After new loves. My mother, dying, gave it to me,
And asked me, when my fate would have me marry,
To give it to my wife.
Iago used the handkerchief to create jealousy between Othello and Desdemona. Iago planted the handkerchief in Cassio's home. This made Desdemona appear unfaithful.
The handkerchief signifies that Desdemona's has been unfaithful when it is found on Cassio. Iago was very clever in choosing to get the handkerchief away from Desdemona.
One must admit that Cassio and Desdemona seem guilty. How else could Cassio have gotten the handkerchief? Iago is a clever, evil person. He will stop at nothing to ruin the lives of those around him. The handkerchief just made it easier for Iago to make Desdemona and Cassio appear guilty. That is why Iago repeatedly asked Emilia to get him the handkerchief. Iago's plan worked and costs Desdemona her life. The handkerchief is significant in that Desdemona did not protect it. She carelessly let it drop and Emilia picked it up and gave it to Iago.