Let $f(r)=\mathcal{H}^{n}(L\cap B(r))>0$, then $f$ is increasing in $r$, and for a.e. $0<r<R$, by the coarea formula,

$$f^{\prime}(r)=\int_{\partial B(r)\cap L}\frac{1}{|\nabla r|}d\mathcal{H}^{n-1}\geq C^{-1}\mathcal{H}^{n-1}(\partial B(r)\cap L)\geq C^{-1}f(r)^{(n-1)/n}.$$

The last inequality is the isoperimetric inequality. Thus $\frac{d}{dr}f^{1/n}\geq C^{-1},$ whence we have the volume lower bound $f(r)^{1/n}\geq C^{-1}r$. ∎