Given a line bundle $L$ on a scheme $X$, we can construct the sheaf of $O_X$ algebras $\bigoplus\limits_{n=0}^\infty L^n$, which by the global Spec functor induces a map from some other scheme to $X$.
How should one think about this process geometrically? I think I'm getting a bit confused from the $O_X(1)$ projective example, since here the scheme obtained is isomorphic to $X$, and this just messes with the ring that we take Proj of (I think).
Are there any illustrative examples of this process where the induced map is not an isomorphism, to illuminate whats going on?
Geometrically $$ \mathrm{Spec}_X\left( \bigoplus_{n=0}^\infty L^n \right) = \mathrm{Tot}(L^{-1}), $$ the total space of the line bundle $L^{-1}$. This is a relative 1-dimensional version of the general fact, that the algebra of polynomial functions on a vector space $V$ is equal to $$ \bigoplus_{n=0}^\infty \mathrm{Sym}^nV^\vee. $$ Dualization in the second formula is responsible for taking the inverse in the first.