Note that the integration region is a triangular region with vertices at $(a,a)$, $(a,x)$, and $(x,x)$ in the $s-\xi$-plane.
Thus, if the inner integral is on $s$, we see that for any fixed $\xi$, $s$ begins at $a$ and ends at $\xi$.
If the inner integral is on $\xi$, we see that for any fixed $s$, $\xi$ begins at $s$ and ends at $x$.
Therefore, we can write
$$\begin{align} \int_a^x \int_a^\xi f(s)\,ds\,d\xi&=\int_a^x \int_s^x f(s)\,d\xi\,ds\\\\\\\ &=\int_a^x f(s)\left(\int_s^x (1)\,d\xi\right)\,ds\\\\\\\ &=\int_a^x f(s)(x-s)\,ds \end{align}$$
as was to be shown!