Intuitive explanation of Nakayama's Lemma

Suppose your module is of finite length. Then you can consider on it the so called radical filtration, which organises the module into an onion-like thing, with elements of the maximal ideal pushing elements of the module farther in from their starting layer to one right below and, moreover, each layer obtained from the one above it in this way.

Now, the condition $\mathfrak m M=M$ tells you that the outermost layer of the module is actually empty: obviously, then, there is not much in the whole thing and $M=0$. We have just discovered Nakayama's lemma!

If your module is arbitrary, exactly the same happens.

Here's something that might or might not make sense to you. You know that every ideal $I$ of a commutative ring $R$ gives rise to an $R$-module $R/I$; these are precisely the $R$-modules on one generator. The $R$-modules of the form $R/m$ where $m$ is maximal are special among these. In this language, the elements of the Jacobson radical are precisely the ones that act trivially on all $R$-modules of the form $R/m$. It follows, for example, that if $I$ is in the Jacobson radical, then we cannot have $IM = M$ for any module of the form $R/m^k$.

Nakayama's lemma asserts that a similar statement is true for all finitely generated $R$-modules. This should be reasonable if you are aware of, for example, the classification of finitely generated $R$-modules when $R$ is a PID. The Wikipedia describes a geometric interpretation of this, but I'm not familiar enough with it to say more.

This is probably not the answer you're looking for, but here's another proof:

Suppose $M \neq 0$ and let $u_1,\ldots,u_n$ be a minimal set of generators for $M$. Then $u_n \in IM$ so we may write $u_n=a_1u_1+\ldots+a_nu_n$ with $a_i \in I$. Hence $(1-a_n)u_n=a_1u_1+\ldots+a_{n-1}u_{n-1}$. Since $I$ was contained in the Jacobson radical, it follows that $1-a_n$ is a unit. Hence $u_n$ belongs to the submodule generated by the $u_1,\ldots,u_{n-1}$, contradiction.