How is Feste able to win over Olivia? How would you track his train of thought in their conversation?
The only place I can think of in this play where you can say that Feste (the Clown) wins Olivia over is in Act I, Scene 5. Is that the place you're talking about?
In that scene, she comes in and he tries to persuade her that she is a fool. He is able to win her over (at least to the extent that she sort of defends him against Malvolio) by his logic.
He proves that Olivia is a fool by talking about the fact that she's mourning her dead brother. He says that if (as Olivia claims) her brother is in heaven, she is a fool to be sad for him.
Is that the part you are talking about or are your referring to some other part of the play?
In William Shakespeare’s play "The Twelfth Night" the play is a pun of fools who torment their adversary the killjoy Malvolio. Through-out the play the fools engage in mischief. Feste is Olivia's jester.
I have to agree with the previous writer that the only place in the play where I can see Feste winning over Olivia is when she tells him to go away. Feste argues with her that he is a good fool. Olivia denies this saying he is basically dull. Feste says that drink will make him more witty. He then makes a witty comment. Oliva asks Malvolio his opinion and the man puts Feste down. Olivia defends Feste and tells Malvolio that he instead is
"Oh, you are sick of self-love, Malvolio, and taste with a distempered "
I would have to say that Feste wins Olivia over with his sense of humor and by winning her support and defence from Malvolilo.