What happened during Viola's interview with Olivia in Twelfth Night?

Olivia’s basic premise is that Orsino doesn’t really know her and has no reason to pursue her.  She tells Viola that she only got in because she was “saucy” at the gate and Olivia was curious.

Viola is able to get in the door by being stubborn.  Olivia is not seeing anyone, because she is in mourning for her father and brother.  Viola will not take “no” for an answer though, and Olivia wants to know more about this unusual young man at her gate.

Viola is pretending to be Cesario, a young man, and she is in the employ of Orsino.  Orsino is in love with Olivia, but she won’t see him, so he sends Cesario/Viola.  Viola’s cleverness gets her into the house.  However, then she has to match wits with Olivia.

Viola tries to start the speech she has prepared, based on Orsino’s script, but she has trouble at first figuring out which lady is Olivia, and then Olivia is veiled.  Viola asks her to show her face.

OLIVIA

Have you any commission from your lord to negotiate
with my face? You are now out of your text: but
we will draw the curtain and show you the picture.
Look you, sir, such a one I was this present: is't
not well done? (Act 1, Scene 5)

The conversation continues along this vein.  When Viola tells her she is beautiful and should not be keeping herself from the world, Olivia says that she has no problem with her beauty, and will “give out divers schedules” of it.  Viola accuses her of being proud.  Olivia explains that Orsino can’t really love her without knowing her.

OLIVIA

Your lord does know my mind; I cannot love him:
Yet I suppose him virtuous, know him noble,
Of great estate, of fresh and stainless youth;
In voices well divulged, free, learn'd and valiant;
And in dimension and the shape of nature
A gracious person: but yet I cannot love him;
He might have took his answer long ago. (Act 1, Scene 5) 

In the end, of course, Olivia falls in love with Cesario instead of Orsino.  She enjoys the matching of wits.  She surprises even herself by falling for him.  Viola doesn’t realize this until Malvolio brings her the ring, a ploy to get Cesario to come back.  Viola then realizes that Olivia loves her as Cesario, and she feels sorry for her, even though she loves Orsino.