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 $\{\sum_{k=1}^n t_k \le \pi\}$, 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 $\{\sum_1^n t_k \le 1\}$ has volume $\frac{1}{n!}$ is discussed [here][1]. 


  [1]: https://math.stackexchange.com/questions/1718021/intuition-for-volume-of-a-simplex-being-frac-1n