I came to your question after looking for the same proof and not finding it anywhere. Following StijnDeVuyst suggestion, I managed to prove it using integration by parts. I will omit the integration limits, just keep in mind they exist.

$$\int \int uv \ dC(u, v) = \int\int uv C'(u,v) \ dudv,$$ where $C'(u,v) = \frac{\partial^2}{\partial u \partial v}C(u,v)$.

\begin{equation}
\tag{1}
\label{main}
\int\int uv C'(u,v) \ dudv = \int v \left(\int u C'(u,v) du \right) dv.
\end{equation}

Solving the inner integral
\begin{align*}
\int_0^1 u C'(u,v) du &= u\frac{\partial}{\partial v}C(u,v) \big\rvert_{u=0}^1 - \int_0^1 \frac{\partial}{\partial v} C(u,v) \ du\\
&= 1-\int_0^1 \frac{\partial}{\partial v} C(u,v) \ du\\
\end{align*}

where the first equality is integration by parts on $u$ and the last equality comes from the fact that $C(1, v) = v$.

Substituting in \ref{main} we have
\begin{align*}
\int \int uvC'(u,v) \ dudv &= \int v \left(1-\int \frac{\partial}{\partial v} C(u,v) \ du \right)dv\\
&= \int_0^1 v \ dv - \int\int v \frac{\partial}{\partial v}C(u,v) \ dudv\\
&= \frac{1}{2}-\int\left(\int v \frac{\partial}{\partial v} C(u,v) \ dv \right) du. \tag{2}\label{segunda}
\end{align*}

Solving the inner integral with integration by parts on $v$ we get
$$ \int_0^1 v \frac{\partial}{\partial v} C(u,v) \ dv = u - \int_0^1C(u,v) \ dv$$

and substituting in \ref{segunda} we have (now including the integration limits)
\begin{align*}
\int_0^1\int_0^1 uv \ dC(u,v) &= \int_0^1 \int_0^1 uvC'(u,v) \ dudv\\
 &= \frac{1}{2} - \int_0^1 u \ du + \int_0^1 \int_0^1 C(u,v) \ dv du\\
&= \int_0^1 \int_0^1 C(u,v) \ du dv
\end{align*}

which proves the identity
\begin{align*}
\rho_S &= 12\int_0^1 \int_0^1 uv \ dC(u,v) - 3\\
&= 12\int_0^1\int_0^1 C(u,v) \ dudv - 3.
\end{align*}

This proof assumes that the copula $C$ has density given by $\frac{\partial^2}{\partial u\partial v}C(u,v)$. When this is not the case, check **Theorem 5.1.1** and **Corollary 5.1.2** from [An Introduction to Copulas by Roger B. Nelsen](https://books.google.com.mx/books/about/An_Introduction_to_Copulas.html?id=B3ONT5rBv0wC&redir_esc=y).