The left-hand side of equation (6.7) only depends on the structure of toric line bundle of $L$ and not on a particular choice of toric section, while the right-hand side seems to depend on the section $s$. We can see directly that the right hand side actually does not depend on the section. If we pick a different toric section, say $s^{\prime}$, then the corresponding support function $\Psi^{\prime}$ differs from $\Psi$ by a linear functional. The polytope $\Delta_{\Psi^{\prime}}$ is the translated of $\Delta_{\Psi^{\prime}}$ by the corresponding element of $M$. The function $\psi_{{\overline{L}},s^{\prime}}$ differs from $\psi_{{\overline{L}},s}$ by the same linear functional and $\psi_{{\overline{L}},s^{\prime}}^{\vee}$ is the translated of $\psi_{{\overline{L}},s}^{\vee}$ by the same element of $M$. Thus the integral on the right has the same value whether we use the section $s$ of the section $s^{\prime}$.