Clarisse is the opposite of Millie--she is open-eyed and observant. She asks questions. She probes and wonders, and THINKS for herself. She is a breath of fresh air for Montag, but she is also dangerous to be associated with since she is everything the government would not approve of and will have to extinguish (which eventually they do).
Clarisse is responsible for Montag's metamorphosis. She gets him to really see the moon and to consider how things work and effect you--smells and tastes, and life in general. She is the stuff that books are made of...in the flesh. Her family is responsible for this since they are all thinkers and doers of actions contrary to the government's approval--they are pedestrians and rememberers of the way things used to be.
Without Clarisse, Montag would still be married to witless Millie, watching the walls of their living room and burning books for a living.