It's enough to check in even dimension, that is for the unit ball $B_n$ in $\mathbb{C}^n$. The map $z\mapsto \pi |z|^2$ from $\mathbb{C}$ to $[0, \infty)$ is measure preserving, and so is the map $\mathbb{C}^n \to [0, \infty)^n$, $(z_1, \ldots, z_n) \mapsto (\pi |z_1|^2, \ldots, \pi|z_n|^2)$. The product of disks $D^n$ maps to $[0,\pi]^n$, and the unit ball maps to the simplex $\{\sum_{k=1}^n x_k \le \pi\}$, which is $\frac{1}{n!}$ of the cube $[0,\pi]^n$. So that's why
$$\operatorname{ Vol}(B_n) = \frac{\pi^n}{n!}$$
Obs: why the simplex $\{\sum_1^n x_k \le 1\}$ has volume $\frac{1}{n!}$ is discussed here.