I can certainly see why you're confused! It looks like you've looked at the wrong problem (or perhaps the right problem, but the wrong problem's answer). That double integral is what would come from a similar situation, but with your plane having equation $8x+9y+z=72$ instead of $8x+7y+z=56$.
To get the integrand, we can solve $8x+7y+z=56$ for $z$ (we needn't do anything else, since the volume is bounded by the $xy$-plane, rather than some more complicated surface). For the inner integral, we start with the equation $8x+7y+z=56$, then set $z=0$ and solve the resulting linear equation for $y$ to give us the upper limit (the lower, $y=0$, comes from the fact that the volume is bounded by the $xz$-plane). For the outer integral, we start with the linear equation we got in the previous step, then set $y=0$ and solve the resulting equation for $x$ to get our upper limit (again, the lower limit comes from the coordinate plane bounding).