Montag recognizes himself in Clarisse's eyes, indicating that there is a reflection of himself in her. Although he doesn't fully realize or recognize what qualities of himself are present in her, he is aware that there is a kinship between them. Clarisse's presence in his life leads him to develop thoughts and feelings that eventually dictate the "radical" actions he takes. Just as Clarisse is different from the norm in Montag's society, he also becomes different. His true feelings, which have gone unheeded, are reflected in Clarisse.
This dystopia that Bradbury has created is built upon primal pleasures. These pleasures are individual, and personal connections are not encouraged. Personal relationships might lead people to want more than just 'pleasure', thus causing citizens to rebel against their society. When Montag sees himself in Clarisse's eyes, this indicates a connection between two souls. The eyes are, after all, the window to the soul. To touch into the humanity is to reject the teachings of this new American society. This foreshadows Montag's rebellion.