A connection $\nabla$ in a vector bundle $E$ over manifold $M$ is a (globally defined) map $$ \nabla : \Gamma(E) \to \Gamma(T^* M) \otimes \Gamma(E) $$ that satisfies the Leibniz rule: $$ \nabla(f \, s) = \mathrm{d}f \otimes s + f \nabla s $$
I often heard that in vector bundles, a connection gives us a way to identify nearby fibers, so we can differentiate sections, however I don't how this works using the above definition. Could any one please explain how a connection allows us to identify nearby fibers and to differentiate section ?
Your help would be greatly appreciated!