Emilia gave the handkerchief to Iago for the simplest of reasons: she didn't realize how important it was going to be, and he was her husband (and she wanted to please him). However, this little piece of cloth became very important. Othello had originally given the handkerchief to Desdemona. Iago convinces Othello that Cassio gave it to Bianca, which convinces him that Cassio and Desdemona are having an affair. When the handkerchief's role in things is revealed in the final act, Emilia realizes all the manipulation Iago's done, and what a monster she's married to.
In Othello, Iago is manipulative and uncaring. He is self-serving and is happy to cause distress to anyone if it furthers his cause to "serve my turn upon" Othello (I.i.42). Iago, whilst revealing to Roderigo that "I am not what I am" (66) becomes Othello's trusted "ancient" and all that Othello sees or thinks he sees strengthens the "ocular proof" which Othello ironically demands in Act III, scene iii, line 364. Othello will reject all other sensible options and rational reasons why Desdemona may have lost the handkerchief. He won't even listen to her as she tells him that Cassio must have found it. He fails to recognize her innocence because of his trust in "honest Iago" which confirms that one of the main themes is the conflict between appearance and reality.
Emilia is also entrapped by Iago although she does recognize that he is "wayward" (III.iii.296). She admits that he has asked her to steal the handkerchief many times which she would never do, especially as she knows how much it means to Desdemona but as Desdemona has just dropped it, Emilia realizes that she can now please her husband and not feel guilty. Emilia does not know the extent of Iago's deceit and is content to "please his fantasy" (303). She gives him the handkerchief because she knows it holds some significance for him and is prepared to allow him that indulgence. Emilia's conflict becomes apparent when the audience sees her loyalty to Desdemona and her outspokenness when she realizes what Iago has used the handkerchief for. Emilia refers to Iago's "pernicious soul" (V.ii.159) and shouts out that Othello has committed "murder! murder!" (170).