Let $P$ be a 2-plane in $\mathbb R^{n+1}$ and $\gamma=P\cap M$. Choose a parametrisation of $\gamma$ by arc-length. Then the curvature of $\gamma$ is $K=||\nabla_{\dot\gamma}\dot\gamma||$, where $\nabla$ is the flat covariant derivative in $P$ (or in $\mathbb R^{n+1}$ as $P$ is totally geodesic). Now, the Levi-Civita covariant derivative $\nabla^g$ of $(M,g)$ is related to $\nabla$ by
$$\nabla_XY=\nabla^g_XY+h(X,Y)\xi,$$
where $h$ is the second fundamental form and $\xi$ is a unit normal vector field. Using your assumption $h(X,X)\ge \delta^2 g(X,X)$ for every $X$ tangent to $M$, one gets
$$K=||\nabla_{\dot\gamma}\dot\gamma||=\sqrt{||\nabla^g_{\dot\gamma}\dot\gamma||^2+h(\dot\gamma,\dot\gamma)}\ge\sqrt{h(\dot\gamma,\dot\gamma)}\ge \delta.$$
This proves your claim.