The geometric interpretation is quite simple: there is a unique torus fiber $L\subset M$ of the moment map $\mu:M\rightarrow\Delta_M$ which is monotone, and this fiber lies over the unique integral point. So this Lagrangian $L$ (equipped with its Spin structure) actually defines an object of the monotone Fukaya category $\mathcal{F}(M)$.
There should be infinitely many monotone Lagrangian tori in $M$ which are mutually not Hamiltonian isotopic (in the two-dimensional case, this is proved by Vianna for general del Pezzo surfaces). However, $L$ is the only one which appears as a fiber of the standard toric fibration.
It can also be shown that $L$ is non-displaceable, and in fact admits non-trivial Floer cohomology $\mathit{HF}^\ast(L,L)\neq0$, so it defines a non-trivial object in $\mathcal{F}(M)$. This is in fact the Lagrangian we use to recover the whole $\mathcal{F}(M)$ (defined over a field $\mathbb{K}$ with $\mathrm{char}(\mathbb{K})\neq2$). One can actually show that by taking iterated mapping cones in terms of $L$ (equipped with suitable local systems) and splitting off direct summands, one can obtain every oriented monotone Lagrangian submanifold in $M$ with non-trivial Floer cohomology. This is the work of Abouzaid-Fukaya-Oh-Ohta-Ono.
In conclusion, you can interpret the existence of a unique integral point as revealing the fact that all the geometric information about (oriented, monotone) Lagrangian submanifolds in $M$ is contained in a unique Lagrangian submanifold $L$. As a simple corollary, we have the following:
For every oriented monotone Lagrangian submanifold $K\subset M$ with $\mathit{HF}^\ast(K,K)\neq0$, we have $K\cap L\neq\emptyset$.
Addendum Let me remark that this pretty simple geometric picture is a consequence of both the torus symmetry and monotonicity. If monotonicity is not assumed, every blow-up (even a very small toric blow-up) on $M$ will create an additional non-displaceable Lagrangian torus, which has been proved recently by Woodward.