Given a commutative ring one may look at it geometrically in terms of its affine scheme - a locally ringed space associated to it, on which its elements behave similarly to functions.
I've read here and there that modules may be thought of as generalized functions over the spectrum. What is the formal justification for this and why, intuitively, should it make sense?
The most basic operation you can hope to perform on a function is to evaluate it at a point. If $m \in M$ is an element of an $R$-module, you can think of its "evaluation" at a prime ideal $P$ as being the image of $m$ in the quotient $M/P = M \otimes_R R/P$ (in the same way as you think of the "evaluation" of $r \in R$ at $P$ as being its image in the quotient $R/P$).
More generally, you can hope to pull back functions along maps, and if $f : R \to S$ is a map of commutative rings inducing a map $\text{Spec } f : \text{Spec } S \to \text{Spec } R$ on spectra, then the "pullback" of $m \in M$ along this map is its image in $M \otimes_R S$. (Note that in order to evaluate elements of modules and pull them back along maps, we first need to evaluate modules and pull modules back along maps. This is the microcosm principle at work.)
An important motivational theorem here is the Serre-Swan theorem, which suggests that if $M$ is finitely generated projective then one should think of $m \in M$ as sections of "vector bundles" over $\text{Spec } R$.