The final thing Clarisse asks Montag is if he is happy. This sets off a chain reaction, particularly in Montag's own mind. She has instigated a conflict in the story--Montag versus himself--and the initiation of this conflict causes another: Montag versus society. That same evening, earlier in the encounter, Clarisse explains the true purpose of firemen, detailing their history of putting out fires rather than setting them. She broadens Montag's horizons here, and questioning his happiness makes him question it as well, something he has never done before. In so doing, he begins to question his path and what he perpetrates daily, which leads to his rebellion. In challenging him to think, rather than simply dismissing what he does not understand and accepting social opinion, she compels him to a different way of perceiving his life and occupation. This is particularly compelling because she is innocent and genuinely curious, rather than malicious, in her questioning.
The last question Clarisse asks Montag the first time they meet in Ray Bradbury's Fahrenheit 451 is "Are you happy?". This question is extremely important to the plot because the idea that Montag is not doing what brings him personal satisfaction and happiness is what makes him look closer at his life: at his job, his wife, and the norms of the society in which he lives.
At first Montag laughs at the idea that he is not happy, but then he looks at the way his wife has separated herself from her life, and how distanced they have become. He cannot even remember where they met.
On the job, Montag is horrified when the firemen burn the house of a woman who not only refuses to leave her books, but starts the fire that will end her life. He becomes confused. When Beatty becomes aggressive and accusatory, Montag has a wake-up call. We learn that he has kept a couple of books hidden himself; he reads one of his books to Mildred and her friends; and, Mildred turns him in and Beatty arrives outside his door, to burn Montag's house.
As Montag looks closely at the world around him, as Clarisse taught him to do, he realizes that he cannot live within the bounds of society and he chooses to run. Across the river, he becomes a part of the book-readers who will rebuild society with books they have read and/or memorized—in a place where the realization of true happiness is possible.
This is probably the first time in his life that Montag has questioned anything. He simply acceptes things. This is the conflict that arises in the book and the catalyst of everything that happens after that