When Bradbury describes Clarisse's "dark" eyes as ones that "were so fixed to the world that no movement escapes them," it is reflective of her character. Clarisse observes everything around her. She takes in the world, using all of her senses to better understand it. She is attune to the world around her. She is not lulled into a false sense of security or a dullness regarding the world and her place in it. The mere asking of the question to Montage of whether he is happy demands him to observe his own life and what he does in order to assess his contentment. In asking Montag to observe, Clarisse wishes for him to be more attune to his world. Clarisse observes something within Montag that makes him fundamentally different from others.
Clarisse's own insistence on observing the world around her is what compels her to ask Montag to do the same. For Clarisse, the need for Montag to observe what is around him is a reflection of how she views the world. Her insistence to Montag is a way for him to appropriate the world through her own eyes. This becomes a reason why Montag finds himself fundamentally changed as a result of his interactions with Clarisse.