It makes certain kinds of representation theory easier to deal with. Rather than just group homomorphisms to automorphisms of a vector space (or ring homomorphi

It makes certain kinds of representation theory easier to deal with. Rather than just group homomorphisms to automorphisms of a vector space (or ring homomorphisms to endomorphisms of a vector space) you can more generally study functors from a category to Vect, the category of vector spaces. This point of view has proven useful in the study of knot invariants like the Jones polynomial. In this case, there is a category of framed tangles and a functor to something called the representation category of quantum sl2, which is a fancy kind of category of vector spaces.