So my lecturer proves the mean value property which poses that under a few conditions on $u$ and
$\overline{B_r(x_0)} \subset \Omega \subset \mathbb{R}^n$:
$$u(x_0) = \dfrac{1}{\text{Vol}(\partial B_r(x_0))}\oint_{\partial B_r(x_0)} u(y)d\sigma(y)$$
In the first part of the proof he writes a coordinate transformation: $y \mapsto x_0 + z \mapsto x_0 + r\zeta$ (because he wants to write the integral in question in a unit disk) which corresponds to an integral transformation:
$$\oint_{\partial B_r(x_0)}u(y)d\sigma(y) = \oint_{\partial B_r(0)}u(x_0 + z)d\sigma(z) = r^{n-1}\oint_{\partial B_1(0)}u(x_0 + r\zeta)d\sigma(\zeta)$$
The first question I have is where does this $r^{n-1}$ come from exactly and how?
I am struggling to understand the concept of 'Volume' in terms of these boundary surfaces which may be related to this; for instance, I don't know why my lecturer writes $\text{Vol}(\partial B_r(x_0)) = \sigma_{n-1}r^{n-1}$ later on in the proof either - where does the $n-1$ power come from? I see that the boundary exists in $n-1$ dimensions, but don't get what this means exactly, if that makes sense?
This also brings me onto his latter, much shorter, proof of the 'Bulk Mean Value Property', namely that now having proved the mean value property, he can say: $$u(x_0) = \dfrac{1}{\text{Vol}(B_r(x_0))}\int_{B_r(x_0)}u(x)dx$$ Here, he starts by saying: $$\int_{B_r(x_0)}u(x)dx = \int_0^r ds\oint_{\partial B_s(x_0)}u(y)d\sigma(y)$$ How are they equal though? I don't understand that at all.
Thanks for any help anyone can provide! I'm really lost.
The factor $r^{n-1}$ comes from the Jacobian of the transformation: see Wikipedia.
As to your second question, you need to apply Fubini's theorem to interpret the n-tuple integral $\int_{B_r(x_0)}$ as an iterated integral.