I discussed this with Rocco Chirivi' and the easiest explanation for us is the following. Writing $x=(x', x_{n-1}, x_n)$, the unit sphere $\mathbb S^{n-1}$ is a "surface" of revolution with the respect to the $x'$ plane and we parametrize as $x'=x',\  x_{n-1}=f(x') \sin \theta,\  x_n=f(x') \cos \theta$ with $x \in B_{n-2}$ and $f(x')=\sqrt{1-\|x'\|^2}$. 

By Guldino's theorem its area element is $d\sigma=f(x') \sqrt{1+\|\nabla f(x')\|^2} dx' d\theta=dx'd \theta$.

Integration in spherical coordinates also yields the result but perhaps in a more obscure way.