There is a combinatorial proof. If $A(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!}$ is an exponential generating function for the number of ways $a_n$ to put a certain structure on a set of size $n$, where $a_0 = 0$, then $\exp A(x) = \sum_{n \ge 0} b_n \frac{x^n}{n!}$ has coefficients $b_n$ which count the number of ways to split a set of size $n$ into subsets, each of which is given an $A$-structure. For example, $A(x) = e^x - 1$ is the structure of "being a nonempty set," and $\exp A(x)$ counts the number of ways to partition a set into nonempty subsets.
Now $\log \frac{1}{1 - x} = \sum_{n \ge 1} \frac{x^n}{n}$ can be thought of as the structure of "being a cycle," which is to say there are $(n-1)!$ ways to arrange $n$ objects into a cycle $s_1 \to s_2 \to ... \to s_n \to s_1$. So $\exp \log \frac{1}{1 - x}$ counts the number of ways to split $n$ objects up into disjoint cycles, which is the same as the number of permutations of $n$ by cycle decomposition. So
$$\exp \log \frac{1}{1 - x} = \frac{1}{1 - x}.$$
A generalization of this argument lets you compute the cycle index polynomials of the symmetric groups. Note that all of the above manipulations take place in the ring of formal power series over $\mathbb{Z}$.