The formula for conditional probability is actually taken as somewhat of an axiom in probability and isn't really derived. It makes sense if you think of conditioning as reducing the sample space of an experiment, and remember that probabilities are always measured in relation to the total sample space. In particular they're always proportions of the total sample space.
When we reduce the sample space to $B$ then the only way for $A$ to occur is if $B$ does as well, hence the $P(A \cap B)$. And the size of this event with respect to the new sample space is $P(A \cap B) / P(B)$, or the fraction of $B$ taken up by $A \cap B$.
It can also help to imagine a Venn diagram when thinking about conditional probability. Whenever you condition on an event you restrict yourself to being in one of the circles, and all conditional probabilities are measured as fractions of the area of that circle.