I always liked the following reason:
Let's call a topology on a space "admissible" if the evaluation function $e: Hom(X,Y) \times X \rightarrow Y$ is continuous. Then the compact-open topology is coarser than any other admissible topology. In particular, in any case where the compact-open topology is admissible, it is the smallest possible topology that does this.
EDIT: See comments for some references. I don't claim any originality here :)