In this passage, Emilia is acting for her husband Iago.  Previously in the scene Desdemona has taken out her handkerchief in an attempt to bind Othello’s head and alleviate his headache; the cloth is too small,  and Othello brushes it away impatiently and it falls to the floor.  In the passage quoted here Emilia has retrieved it, and is speaking of her resolve to give it to Iago, for he has often begged her to steal it from Desdemona (Emilia is Desdemona’s attendant).  This had proved impossible, however, for Desdemona kept it about her person at all times, for it was the first gift Othello ever bestowed upon her, and therefore she values it greatly.  Emilia states that she will “have the work ta’en out,” which here means that she will have the embroidery copied, and give the piece to Iago, who plans to plant the handkerchief in Cassio’s quarters, therefore condemning him and Othello’s young bride both.  Emilia, however, knows nothing of these plans and desires only to please her husband.

In this section of the play Iago has already planted a seed of doubt in Othello’s mind as to the actions of Cassio and Desdemona, hence Othello’s shortness of temper when Desdemonda comes to him with the handkerchief.  Shortly after the quoted passage takes place and Emilia alerts her husband to her actions, Othello returns and speaks in anguish of the suspicion brewing in his mind and the unrest it causes.  He demands to Iago:

Make me see’t; or at the least so prove it
That the probation bear no hinge nor loop
To hang a doubt on – or woe upon thy life!

Thus Othello has demanded concrete proof that his wife is being unfaithful – proof that Iago has now acquired in the form of Desdemona’s dropped handkerchief.  Finally, Othello’s wrath is being further provoked with the loss of this token, for in this passage Emilia notes that Othello “conjur’d her, she should ever keep it,” that is, he told Desdemona to guard it always, and never let it escape her possession.  That he should assume she has given it to Cassio willingly, Iago knows, will send him into a blind, jealous rage, and wreak vengeance upon them both, thus demoting Cassio from his rank as lieutenant and doubtlessly promoting the “friend” Iago to this position.