Why is it appropriate that the story says of Montag "He felt his body divide itself..." in Fahrenheit 451?

I think that you could argue that this is the point in the
story where Montag is, figuratively, becoming two different people.  If so, then this is
a very appropriate thing for the story to say.

For most of
his life, Guy has been one person -- he is a fireman.  As such, he stands for the values
of his society.  But now, his experiences with Clarisse and the near-death of his wife
Millie have started to make a new him.  Because of these things, he has started to
question the values of the society.

When he does, he starts
to become two people.  The two parts of him (just as this passage says) become different
from one another and they start to be "two halves grinding" against one another.  He is
starting to have different parts of him that are coming in conflict with one
another.