Suppose ${\bf x}'$ does not lie on $n$ linearly independent binding hyperplanes. (Remember that a hyperplane in $\mathbb{R}^n$ is a linear equation in $n$ variables.) Then the maximum number of linearly independent hyperplanes binding at ${\bf x}'$ is $r < n$. Let $G$ be the $r \times n$ matrix of constraint coefficients, and let $G{\bf x}' = {\bf g}$. Since the rank of $G$ is $r$, the nullspace of $G$ has dimension $n-r > 0$. Thus there exists some ${\bf d} \neq {\bf 0}$ such that $G{\bf d} = {\bf 0}$. Therefore $G({\bf x}' + {\bf d}) = G({\bf x}' - {\bf d}) = {\bf g}$, which means that both ${\bf d}$ and $-{\bf d}$ are feasible directions at ${\bf x}'$ (i.e., for a sufficiently small distance from ${\mathbf x}'$ in these directions we're still in $P$, as the only constraint hyperplanes to worry about sufficiently close to ${\bf x}'$ are those that are binding at ${\bf x}'$). Then there exists $\epsilon > 0$ such that ${\bf x}_1 = {\bf x}' + \epsilon {\bf d}$ and ${\bf x}_2 = {\bf x}' - \epsilon {\bf d}$ are both in $P$. Since ${\bf x}' = \frac{1}{2} {\bf x}_1 + \frac{1}{2} {\bf x}_2$, ${\bf x}'$ cannot be an extreme point.
Thus if ${\bf x}'$ is an extreme point, it must lie on $n$ linearly independent binding hyperplanes.