Her parting question is one that changes his entire perspective on his life. She asks, as they are parting after their first meeting, "Are you happy?" At first, he laughs the question off and thinks to himself, "Of course I'm happy!" Of all of the things to ask. After all, didn't the novel open up with the words "It was a pleasure to burn," and then go on to describe Montag's firece joy at burning books? And then he meets this girl, who is different, who asks probing questions, who seems like a breath of fresh air, and she tosses that question at him right as they part. Montag considers her question "nonsense."
By the end of the evening however, he has to seriously reconsider his initial rejection of her question. As he steps into his cold, empty house, he remember the book he has hidden behind the grille; why would he have that if there wasn't some part of him that wasn't content with life as it was. As he goes into the bedroom and feels the emptiness there too, pretty soon his security in his own happiness fades quickly. He realizes, quite suddenly,
"He was not happy. He was not happy. He said the words to himself. He recognized this as the true state of affairs. He wore his happiness like a mask and the girl had run off across the lawn with the mask."
Clarisse, with her question, has forced himself to face the fact that he wasn't happy at all. He lives an empty life, with a wife that is so miserable that she tries to kill herself, but in total denial over that misery, and he doesn't know where to go from there. Her question launches him into a journey of self-discovery and questioning that will change his life for good. I hope that helps a bit; good luck!