I think geometry provides a good reason: elements of a module over a ring are like generalized functions on its spectrum. We can talk about the support of a module element, or its vanishing set. More concretely, think of how global sections of a line bundle can act as functions mapping your scheme into projective space.
When you glue together a module on open sets of a spectrum or a scheme, you get to glue using maps which are module isomorphisms, which are which are more flexible than the ring isomorphisms required to glue together a scheme. Borrowing intuition from smooth manifold land, the twist in the Moebius band (as a line bundle on the circle) is formed by gluing a copy of the reals to itself via multiplication by -1, a module map, not a ring map, which allows to think of functions like $\cos(\theta/2)$ as being globally defined: as a map to the Moebius band.
Similarly, when you have a representation V of a group G, each element v in V gives you a nice evaluation map from G into V, so again, we're getting morphisms from an object into a known object.