It's easiest to understand for local rings, so let $R$ be one with residue field $k$. Nakayama's lemma just says that a finitely generated $R$-module is zero if and only if the induced $k$-vector space is. Through the magic of abelian categories, this implies that a map of $R$-modules is surjective if and only if the induced $k$-linear map of $k$-vector spaces is (apply the lemma to its cokernel). This says that I can find generators for an $R$-module by lifting a basis of its associated $k$-vector space (that is, I can test whether a map $R^n \to M$ is surjective by testing it after reducing by $k$).
There are two ways to look at this: one (algebraically), it allows you to consider a lot of $R$-module statements as actually being $k$-linear algebra statements; and two (geometrically), it allows you to transfer information from the fiber of a sheaf at a point to the stalk at that point, and from there, to an open neighborhood.
An example of the first property: suppose you want to prove the Cayley-Hamilton theorem for a linear endomorphism $A$ of some finitely-generated $R$-module: that $A$ satisfies its own characteristic polynomial $p_A$. Note that $p_A$, as an element of $R[t]$, reduces correctly when we pass to $k$, so that $p_A(A)$ vanishes after reducing to $k$ by the Cayley-Hamilton theorem for vector spaces. Therefore, by Nakayama's lemma applied to the cokernel image of $p_A(A)$, it vanishes over $R$ as well.
An example of the second property: suppose $R$ is noetherian and I have a flat $R$-module $M$, and I choose a basis for its reduction to $k$, giving a presentation $R^n \to M \to 0$ (it is surjective by the lemma applied to the cokernel, as explained before). This turns into a short exact sequence $0 \to K \to R^n \to M \to 0$ in which $K$ is finitely generated (since $R$ is noetherian) and since $M$ is flat, it remains exact after reducing to $k$, where the kernel $K$ vanishes. Conclusion: $M$ is free over $R$. The geometric interpretation of this is that flat, coherent sheaves over a noetherian scheme (if you're reading Shafarevich, your schemes are varieties and are always noetherian) are vector bundles.