A morphism of stacks $f: X \to Y$ is representable if for every scheme $U$ and morphisms $y:U \to Y$ the fiber product $X \times_{Y,y} U$ is an algebraic space.
A stack is a functor on schemes taking values in groupoids. The $f: U \to Y$ I imagine is actually confusing the functor of points of $U$, $h_U$, with $U$.
However, $h_U$ is a functor taking values in the category of sets.
I can't quite make sense of the morphism $f:U \to Y$ since it is a morphism of functors taking values in two different categories.
How does one make sense of this?
You're precisely right, in that we are conflating $U$ and it's functor of points $h_U$.
You may think of sets as groupoids with no nonidentity morphisms. More precisely, the category of sets embeds fully faithfully into the category of groupoids. Using this, we may view $U$ as a functor valued in groupoids.