$\def\Vol{\operatorname{Vol}}$As Evans does in his proof of the mean value formula, define $\phi\colon [0,r] \to \mathbb R$ by
\[
\phi(s) = \frac 1{\Vol(\partial B_s)} \int_{\partial B_s} u(x)\, dS(x)
= \frac 1{\Vol(\partial B_1)} \int_{\partial B_1} u(sx)\, dS(x)
\]
We have
\begin{align*}
\phi'(s) &= \frac 1{\Vol(\partial B_1)} \int_{\partial B_1} Du(sx)x\, dS(x)\\
&= \frac 1{n\alpha(n)s^{n-1}} \int_{\partial B_s} Du(x)\, \frac xs\, dS(x)\\
&= \frac 1{n\alpha(n)s^{n-1}} \int_{\partial B_s} Du(x)\, \nu_{B_s}(x)\, dS(x)\\
&= \frac 1{n\alpha(n)s^{n-1}} \int_{\partial B_s} \frac{\partial u}{\partial \nu}(x)\,dS(x)\\
&= \frac{1}{n\alpha(n)s^{n-1}} \int_{B_s} \Delta u(x)\, dx\\
&= -\frac 1{n\alpha(n)s^{n-1}}\int_{B_s}f(x)\, dx
\end{align*}
As $\phi$ is differentiable on $(0,r)$ and continuous on $[0,r]$, we have
\begin{align*}
\phi(r) -\phi(0) &= \int_0^r \phi'(s)\, ds\\
&= -\int_0^r \frac 1{n\alpha(n)s^{n-1}} \int_{B_s} f(x)\, dx\; ds\\
&= -\frac 1{n\alpha(n)} \int_0^r s^{1-n}\int_0^s \int_{\partial B_\sigma} f(x)\, dS(x)\; d\sigma\; ds\\
&= \frac 1{n\alpha(n)} \int_0^r \int_\sigma^r s^{1-n}\int_{\partial B_\sigma} f(x)\, dS(x)\; ds\; d\sigma\\
&= -\frac 1{n(n-2)\alpha(n)} \int_0^r \int_{\partial B_\sigma} f(x)\, dS(x)\; d\sigma\\
&= -\frac 1{n(n-2)\alpha(n)}\int_0^r \int_{\partial B_\sigma}\left(\frac 1{\sigma^{n-2}} - \frac 1{r^{n-2}}\right) f(x)\, dS(x)\;d\sigma\\
&= -\frac 1{n(n-2)\alpha(n)}\int_0^r \int_{\partial B_\sigma}\left(\frac 1{|x|^{n-2}} - \frac 1{r^{n-2}}\right) f(x)\, dS(x)\;d\sigma\\
&= -\frac 1{n(n-2)\alpha(n)}\int_{B_r} \left(\frac 1{|x|^{n-2}} - \frac 1{r^{n-2}}\right) f(x)\, dx
\end{align*}
Now, as $\phi(0) = u(0)$ and
\[ \phi(r) = \frac 1{\Vol(\partial B_r)}\int_{\partial B_r} g(x)\, dS(x) \]
the desired result follows.