Montag literally sees himself in Clarisse's eyes and this is significant because she, like a mirror, challenges him to look at himself - not just his reflection but his entire existence and the world he lives in. Clarisse asks him probing questions such as whether or not he is happy and whether he's read any of the books he's burned. Clarisse gets Montag to start questioning everything in his life. She literally and figuratively serves as a kind of interrogating mirror; she challenges Montag to look at himself. He thinks about this following one of their conversations and remarks how rare it is that he encounters someone who interacts significantly with him. Most of his relationships, even with his wife, are based on mindless repetition. But with Clarisse, Montag gets a sense of real connection and a sense of significant communication. He thinks:
How like a mirror, too, her face. Impossible; for how many people did you know that refracted your own light to you? People were more often--he searched for a simile, found one in his work--torches, blazing away until they whiffed out. How rarely did other people's faces take of you and throw back to you your own expression, your own innermost trembling thought?
This scene comes when Montag first meets Clarisse, and she expresses the surprising opinion (to him) that she is not afraid of him, because although most people are scared of firemen, she can see that he is just a man:
He saw himself in her eyes, suspended in two shining drops of bright water, himself dark and tiny, in fine detail, the lines about his mouth, everything there, as if her eyes were two miraculous bits of violet amber that might capture and hold him intact.
(Bradbury, Fahrenheit 451, Google Books)
By seeing himself in her eyes, Montag comes to the realization that she is correct; he is just a man, like any other man. More importantly, he is a product of society, common, ordinary; his job is just another job, no more or less important than that of anyone else. This realization starts the process that shocks him out of his complacency, and gives him the motivation to try and become a reasoning individual instead of a faithful, unthinking cog in society.