The handkerchief was given to Othello by his mother. It is the only remaining link to his mother that he possesses and as such is very valuable to him. So when he falls in love with Desdemona and gives it to her, she knows that it was a very precious gift. Iago sees a way to use it in his plot to drive Othello mad and so begins to let Emilia know that he wants it. Without knowing its great importance to Othello, she removes it from Desdemona's bedroom in order to give it to Iago.
Othello has begun to rage against what seems to be Desdemona's infidelity with Cassio but still demands that he have "ocular" proof or he cannot believe it. In order to produce such proof, Iago brings Othello to witness the exchange between Cassio and Iago when Cassio has the handkerchief and proceeds to joke about how Bianca cannot help but hang on him and he can't get rid of her. Iago has Othello believing that he is discussing Desdemona and this throws Othello into a fit of rage that leads to a fit of epilepsy as well and cements his resolution to kill Desdemona.
Othello returns to Desdemona and is further enraged when she cannot produce the handkerchief, considering this even more proof that she has been unfaithful and has given it to Cassio since she cannot find it in her room.