Any tangent hyperplane is uniquely determined by its normal versor, that via the Gauss map can be identified to a point is $S^n$. So the tangent hyperplanes form a submanifold of dimension at most $n$ in the space of hyperplanes of $\mathbb{R}^{n+1}$. Such a space is  the dual space $(\mathbb{R}^{n+1})^*$, whose dimension is $n+1$, so the result follows.