Montag and Mildred are desperately afraid that they are going to be caught.
When the woman refuses to leave her books, and decides to die with them, Montag starts thinking.
There must be something in books, things we can't imagine, to make a woman stay in a burning house; there must be something there. You don't stay for nothing. (part 1)
When Captain Beatty comes to visit Montag, he explains why he should not try to read the books.
[The] books say nothing! Nothing you can teach or believe. They're about non-existent people, figments of imagination, if they're fiction. And if they're non-fiction, it's worse… (part 1)
Mildred says “Books aren't people” and tells him they are not alive. She is worried that they will get caught, since having books is against the law. She is worried that if they get caught the firemen will come burn the house down, and her television with it.
Montag is worried that there are no copies of these books left in the world, and Mildred is just worried about her friends coming over to watch television.