The idea of the woman dying with her books is an important moment and symbol. It shows that the voice of dissent and active resistance cannot be thoroughly silenced by an external force. More critically, it allows Montag to fully understand that there are some things that define life, and, if taken away, can define how one chooses to die. In a society that does not permit any real and substantive choice on the part of the individual, the old woman poses a great counterresponse. At the moment of encounter, she is able to exercise the only choice that really matters in choosing to take her own life. She proves to Montag that the notion of seeing human choice as variables in an equation that can be balanced out and counteracted is not a way to live life. This form of life might provide convenience and comfort, but it is not the sole path one can accept in attempting to live life. This is where Montag is, and the reason why the woman dying with her books leaves such a profound impact on him.
In Fahrenheit 451, the woman burns herself, presumably, to become a martyr for her cause.
Her suicide raises the stakes. The novel is full of biblical allusions, and the woman going up in flames, as Christian martyrs are famous for doing, fits with the allusions and connects her to Christian martyrs. Her death is a statement, and as far as Montag is concerned, a powerful one. Her death is an old time testimonial. Her death is a catalyst for Montag's transformation. That she cares so much for books she is willing to die for them moves Montag in the direction of beginning a new, thought-filled life.
Standing among her books as she, and them, go up in flames, is an image reminiscent of a martyr being burned at the stake. She lights the match herself to demonstrate her free will, and her power to choose. She willingly gives up her life to testify to the value of what the firemen are destoying. And Bradbury writing her as a martyr for books connects her to the Christian martyr the Captain tells Montag about.
To me, the short answer to this question is that she lights the match because she would rather die as a human being than live as something that is less than human. By saying this, I am saying that the people of the society in this book are not really fully human.
In this society, "normal" people do not think. They do not really have any emotional lives. They go from day to day "entertaining" themselves in mindless ways. I would argue that this is not a truly human existence since what makes us human is our ability to think and to have true relationships with people.
When the firemen come to destroy the old woman's books, what they are really going to destroy is her humanity. They are going to take away her ability to think. If they burn her books, she will be stuck somewhere, maybe watching the parlour walls like Millie Montag does. That is not a fully human life and the old woman would rather die than live like that.
The woman you are referring to is Mrs. Blake. Her act of defiance reflects the importance of literature, the possession of books is denied in the society portrayed in Fahrenheit 451. She opts to stay in the house as an act of refusing what is important to her, her books. She is taking on the role of a Martyr, dying for her freedom to possess literature and refusing to conform. This inspires protagonist, Guy Montag in Fahrenheit 451