This quote about Clarisse signifies the beginning of Montag's journey of self-discovery and escape from the horrible society in which he lives. Clarisse forces Montag to realize that he is different from the other people in his life by questioning him and deliberately "bothering" him. Later on in the book, Montag says to Mildred that people "need to be really bothered once in a while." By this he means that sometimes something has to disturb the status quo in a person's life in order for them to really start thinking. Clarisse disturbs the natural goings-on of Montag's life and makes him begin to question himself.
And when Montag does start looking at himself, he realizes that he is like Clarisse; that he, too, is discontent with the way things are and yearns to know more.
"He saw himself in her eyes, suspended in two shining drops of bright water . . ."
He genuinely looks at her and sees something familiar, which comes as a stark contrast to his wife. Montag and Mildred have nothing in common, a fact that becomes more and more apparent as the story progresses. She and Montag rarely talk about anything serious or substantial, and she frequently refuses to give him her undivided attention for even a moment because she'd rather watch the program on the parlor walls. She even calls the TV characters family, and it is heavily implied that Mildred cares more for her virtual life than for her husband.
Montag is similarly detached from his wife. He is frightfully aware that neither of them remember how they met each other. At one point Clarisse makes him realize that he doesn't love Mildred and never has, but has merely deluded himself into thinking that he is happy. Near the end of the book, Montag comments that he ought to be sad if she dies in the bombing, but he knows he won't be.
When Montag first meets Clarisse, she is vibrant and full of life, whereas when he returns home he finds his wife unconscious and half-dead. Near the end of the book, Montag notes that Mildred's face is "wildly empty." The audience's first encounter with her is when she overdoses and Montag finds her and thinks that her face like "a snow-covered island." A few pages earlier Montag compares most people he knows to fires that will soon be extinguished, and this how he finds Mildred, nearly extinguished entirely.
Clarisse is contrasted heavily with Mildred, which slowly begins to prove to Montag that he isn't like all the empty, vacant people that he knows, but rather that he is like Clarisse. He questions, he's curious, and he is desperate for something more. Long before meeting Clarisse he had been stealing and hiding books for reasons that he was unable to admit to himself. But coming into contact with Clarisse shows Montag his own true nature and, now, having seen it, he is unable to go back to pretending that he isn't different.
Montag is thinking about Clarisse in this quote. She makes him look at himself, like a mirror. When Clarisse asks him questions, such as whether or not he is happy, she makes him consider who he is.
Clarisse is a young woman Montag met briefly. Montag describes Clarisse as like a mirror.
How like a mirror, too, her face. Impossible; for how many people did you know that refracted your own light to you? (Part 1)
In Montag’s world, people have taken for granted how closely everything is controlled. Things are done for them, and they are happy. Clarisse makes Montag question his very soul—his inner being. He has never asked himself if he was happy. He has never thought about being happy. She makes him explore hard truths about his life.