In Othello, how does Bianca's return with the handkerchief help Iago?
Just to add a few more specifics to how the handkerchief furthers Iago's plot:
1. Othello knows that Cassio has been seen around Cyprus with Bianca; so when Bianca shows up with the handkerchief in her possession, she provides the "ocular proof" that Othello asks Iago for. Bianca's having the handkerchief causes Othello think that her lover Cassio obtained the handkerchief from Desdemona and then passed it on to Bianca. Iago knows that he has does enough to taint Othello's view of Cassio and Desdemona, that the conclusion above is the only one that Othello will draw.
2. Bianca is a woman of ill-repute (a prostitute); so when Othello sees her with the handkerchief not only does it give credence to Iago's affair theory, but it also angers him to think that his prized possession was worth no more to his wife than for her to give it to a man who would in turn bestow it upon a prostitute. This enrages Othello because it cheapens his mother's memory and makes him consider her eerie warning about one's losing the handkerchief (marital infidelity).
Given the enormous import given the token by Othello, Iago knows that it will be a very powerful piece of evidence to use in convincing Othello that Desdemona is in fact cheating on him with Cassio. Othello had waxed poetic about the handkerchief previously, saying that he was given it by his mother, that he treasured it more than almost anything else he owned.
By bringing it back to Iago, Bianca had no idea that it would be so valuable to him or what he was going to use it for, but Iago was thrilled, knowing how he could use it to enrage Othello.