This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that people are interested in studying such Galois representations. There are lots of theorems which explain images of such representations are contained in special/open/closed subgroups or, form special subgroups. What geometric information can we get from such theorems? (or, why are such theorems so good for us?)

For example, the criterion of NéronOggShafarevich says that if the Galois representation on the Tate module of an Abelian variety is unramified, the Abelian variety has good reduction. For this and more examples, see "Good Reduction of Abelian Varieties" by JeanPierre Serre and John Tate. Another example is the Tate conjecture for Abelian varieties $A, A'$ over finitely generated fields $K$: $\mathrm{Hom}(A,A')\otimes_{\mathbf{Z}}\mathbf{Z}_\ell = \mathrm{Hom}_{G_K}(T_\ell A,T_\ell A')$ The surjectivity means roughly that you can construct isogenies between Abelian varieties from $G_K$equivariant homomorphisms between their Tate modules. Two of the equivalent formulations of modularity of elliptic curves are: 1) The Galois representation is modular [Galois representation]. 2) There is a finite morphism $X_0(N) \to E$ [geometry]. 

