The statement is true if and only if the feasible region is bounded. Consider the inventory optimization problem
$$\min_{x,y,z} \left\{\sum_t z_t : z_t \geq h y_t, z_t \geq -by_t, y_t = y_{t-1}+x_t-d_t, 0 \leq x_t \leq x_{\max} \right\},$$
with starting inventory $y_0$, inventory level $y_t$, production size $x_t$ (bounded above by some machine limitation $x_{\max}$, expected demand $d_t$, holding costs $h$ and backlogging costs $b$. This problem is feasible and has an optimal solution. The feasible region of this problem is unbounded ($z_t$ can grow infinitely large), so the statement is not true.
When the feasible region is bounded, it is a convex polyhedron whose extreme points are basic feasible solutions.