Emilia, like so many other characters in the play, including Othello and Desdemona, is deceived by Iago. She knows more than some of the others that he is not entirely a good person—she calls him "wayward," which means unpredictable or perverse. However, like the others, she does not realize the depth of his corruption and depravity. Iago is far more than "wayward" or unpredictable—he is evil.
Emilia, precisely because she is a good person who cannot imagine how deeply and wholly her husband wants to destroy Othello, gives Iago the handkerchief after Desdemona drops it. She does this because she wants to please her husband. She explains her thinking in the following speech in act 3, scene 3:
I am glad I have found this napkin,
This was her first remembrance from the Moor.
My wayward husband hath a hundred times
Wooed me to steal it, but she so loves the token
(For he conjured her she should ever keep it)
That she reserves it evermore about her
To kiss and talk to. I’ll have the work ta’en out
And give ʼt to Iago. What he will do with it
Heaven knows, not I.
I nothing but to please his fantasy.
As Emilia explains, Desdemona loves this handkerchief because it was a gift from Othello. Emilia also explains that she doesn't know why Iago wants it. She knows her husband realizes that she loves Desdemona and can't conceive that he would ruthlessly use her, his own wife, to enable his plot to destroy her mistress as "collateral" damage in his real quest to destroy Othello.
Emilia's willingness to indulge her husband reveals how skillfully Iago has hidden his depravity from others. Like many sociopaths, he is able to imitate the behavior of normal people with empathy while being inwardly filled with rage and hate. Emilia has no idea, for example, that he thinks all women, including her, are "whores" and animals and therefore thinks Desdemona's death is justified. She has no conception, until it is far too late, of what he has been plotting. When she does find out, she exposes him.