The Clown says he can prove Olivia is a fool.
The fool has been having a good time with Maria (Olivia's maid), so he is already warmed up by the time Olivia shows up. The exchange shows that Olivia is good-natured.
When the fool first spars with Olivia, he says that he can prove that she is a fool.
Misprision in the highest degree! Lady, cucullus
non facit monachum. That’s as much to say as, I
wear not motley in my brain. Good madonna, give
me leave to prove you a fool. (Act 1, Scene 5)
The first sentence means basically “miscalculation” to the highest degree. He is saying that just because he looks silly does not mean he is silly. He will prove that she is the fool.
Olivia has been mourning her brother, who has died not recently enough for her behavior. Stop moping, he is telling her! She has rejected all of Duke Orsino’s advances on these grounds. The fool uses this as a reason to contradict her.
Good madonna, why mournest thou?
Good fool, for my brother's death.
I think his soul is in hell, madonna.
I know his soul is in heaven, fool.
The more fool, madonna, to mourn for your brother's
soul being in heaven. Take away the fool, gentlemen. (Act 1, Scene 5)
Basically, he is saying that because her brother is in Heaven, which is supposed to be a wonderful place, and not in Hell, which would be terrible, she should not be so upset about it!
Olivia reacts good-naturedly to this, even standing up for the clown when Malvolio becomes annoyed. She points out that it is his job to annoy people! She is a gentle person, not afraid to be put in her place, and generally strong enough to hold her own in an argument.
In Elizabethan times, and Shakespeare’s plays, the fool or clown was often the wisest character, despite his name. Therefore he is used in this play to match wits with various characters, though puns and wordplay. Often, Shakespeare will use him to teach someone a lesson or just to tell a truth. In this play, since there is some misdirection and hidden identity going on, for example, look to the fool to be aware that a certain Viola is indeed a man!