Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$

For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(C,J)\subseteq Fun(C^{op},Set).$$

Part of the beauty/usefulness of this procedure is that the resulting objects are topoi and working in them is "easy".

The first question is: why only $Set$?
Usually when consider functions/morphisms/functors targeting some structure, the set/category of functions/functors inherits this structure. ("Usually" our structure comes from products and limits so "usually" it works. I never worked with Hopf algebras to not use the usually.)
So it may seem that we want to "pullback" set's structure.

The second question is why we do it for other categories as well? We consider sheaves of groups or rings. We will never get a topos, but we are nevertheless interested. And it seems that some people are interested even in sheaves over other less concrete categories. (I don't have an explicit example.)

So the main question is: why do sheaves always seem to pop out? Why it seems that sheaves contain interesting information?