Notice that the vanishing of $\frac{\partial}{\partial x_1} \sigma_{m,r}, \cdots, \frac{\partial}{\partial x_m} \sigma_{m,r}$ implies the vanishing of $\sigma_{m,r}$ since
$$\sigma_{m,r}=\frac{1}{r}\left( x_1\frac{\partial}{\partial x_1} \sigma_{m,r}+ \cdots + x_m\frac{\partial}{\partial x_m} \sigma_{m,r}\right),$$
so we can just focus on the system of equations $\frac{\partial}{\partial x_i}\sigma_{m,r}=0$ as the defining equations for $S_{m,r}$. We can give a proof by induction. Start with $r=2$, where the defining equations become
$$\frac{\partial}{\partial x_i}\sigma_{m,r}=\left(\sum_{k=1}^m x_k\right)-x_i=0$$
therefore $x_i=0$ for all $i$. For larger $r$ we proceed as follows. First we notice that
$$\frac{\partial}{\partial x_i}\sigma_{m,r}=\sigma_{m,r-1}-x_i(\frac{\partial}{\partial x_i}\sigma_{m,r-1}).$$
So the system of equations is equivalent to
$$x_i\frac{\partial}{\partial x_i}\sigma_{m,r-1}=0.$$
Suppose that $q$ of the variables vanish, and the remaining $m-q$ are nonzero. If $m-q\geq r-1$ then the elementary symmetric function are nontrivial, and by induction (use the statement for $m'=m-q, r'=r-1$) we have that $x_i\frac{\partial}{\partial x_i}\sigma_{m,r-1}$ cannot all be zero. Now we can conclude that $m-q\le r-2$ or in other words $q\geq m+2-r$. Of course we can check that this is in fact enough, because if $m+2-r$ variables vanish then every monomial in our equations vanishes.