What is the significance of Montag seeing his reflection in Clarisse's eyes in Farenheit 451?

For one thing, this makes him recall a moment with his mother. When he was a child, the power had gone out and they were left with one candle and found themselves in an intimate moment of person-to-person communication and glances. In this society where people gave all of their attention to technological devices (parlour shows), such interaction would have been rare. Seeing himself in Clarisse's eyes makes him recall this idea of people interacting with one another and actually communicating. 

One time, when he was a child, in a power-failure, his mother had found and lit a last candle and there had been a brief hour of rediscovery, of such illumination that the space lost its vast dimensions and drew comfortably around them, and they, mother and son, alone, transformed, hoping that the power might not come on again too soon . . . 

A few pages later, Montag reflects some more upon the idea of Clarisse as a mirror. He tries to think about why this has affected him so much. He concludes: 

People were more often - her 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? 

In Montag's experience, people do not listen to one another. They simply spew out their own (thoughtless) thoughts and these are usually selfish and/or without any genuine sense of empathy or consideration of the person he/she might be talking to. But Clarisse, being a thoughtful person, is considerate and empathetic. She throws a lot of questions at Montag because she is curious but she also wants to know about him. When she ends their conversation by asking if he is happy, Montag is confounded. He never gets this kind of depth and interest from his wife, Mildred. His conversations with Clarisse cause him to think more deeply about a lot of things and perhaps most significantly, these conversations implore him to evaluate himself. This is the manifestation of his idea that Clarisse was like a mirror because she has provoked him to reflect upon himself.