How does Montag react to the woman who dies with her books?
Being a part of burning the books and home of the woman who refuses to save herself and instead burns with her books is a turning point for Montag in Bradbury's Fahrendeit 451, and the effects are evident immediately.
He is sullen and quiet during the ride back to the station, and can't get the words the woman said out of his mind. When he mentions what she said, the captain explains the words:
'We shall this day light such a candle, by God's grace, in England, as I trust shall never be put out,' said Beatty....A man named Latimer said that to a man named Nicholas Ridley, as they were being burnt alive at Oxford, for heresy, on October 16, 1555.
The quote parallels the woman and her situation, of course, and the candle connects with the fire that destroys her, as well as the fire that Montag sets for a living.
Arriving home that night, Montag is trance-like and the scene in his bedroom is almost surreal:
So it was the hand that started it all. He felt one hand and then the other work his coat free and let it slump to the floor. He held his pants out into an abyss and let them fall into darkness. His hands had been infected, and soon it would be his arms. He could feel the poison working up his wrists and into his elbows and his shoulders, and then the jumpover from shoulder blade to shoulder blade like a spark leaping a gap. His hands were ravenous. And his eyes were beginning to feel hunger, as if they must look at something, anything, everything.
Montag later asks his wife if she remembers when they met. In the next few pages of the narrative he wonders how people get so empty. He asks Mildred about Clarisse. He sees shadows outside and thinks the hound is out there. He tries to tell Mildred about the woman. He takes a book from its hiding place, then decides to stay home from work.
Montag is deeply disturbed by the events at the woman's house, and he will not return to his former state of emptiness.
That lady's death (along with meeting Clarisse and Millie's near death) is what really makes Montag rethink his life. Therefore, it is one of the most important events in the book.
When Montag sees that the woman is willing to die rather than to be parted from her books, it makes a huge impression on him. It really makes him think that there must be something important about books if someone would feel that strongly about them.
So he reacts to her death be having that feeling and then by acting on it -- by looking at the books that he has and reading them.