I think Shakespeare chose to make this a story about the perversion of intimacy and the process of accruing evidence in service of an accusation because it makes for an interesting plot that brings up feelings people can relate to in one way or another. The handkerchief shows that when you’re jealous you really can’t think straight. Othello believes Desdemona is cheating just from the handkerchief being found in Cassio’s possession. To the reader, this can seem a bit ridiculous, it is just a handkerchief. But, Othello, who is already very jealous, sees this as the definite sign that Desdemona is cheating. Iago knows Othello will jump to conclusions if the handkerchief is found with Cassio and uses this to manipulate Othello. When we are jealous, our minds deceive us by seeing the tiniest details as major signs.