I logically understand this theorem, but I don't intuitively understand with picture.
Let $S$ be a nonempty convex open set in $\mathbb R^n$ and let $f\colon S\to\mathbb R$ be differentiable on $S$. Then $f$ is convex if and only if for each $x_1,x_2\in S$ we have $$[\nabla f(x_2)-\nabla f(x_1)]^t(x_2-x_1)\ge 0. $$ Similarly, $f$ is strictly convex if and only if for each distinct $x_1,x_2\in S$ we have $$[\nabla f(x_2)-\nabla f(x_1)]^t(x_2-x_1)> 0. $$
It's telling you that the gradient spreads out in all directions so that the more you move in one direction, the more the gradient points in that same direction.