For a smooth function $f: \mathbb{R}^2 \to \mathbb{R}$, the gradient at a point $(x_0, y_0)$ is perpendicular to the level curve at that point. If $c = f(x_0, y_0)$, the level curve is $f(x,y) = c$. If you move along a level curve, then the value of $f$ does not change at all. If you move perpendicular to the level curve (i.e., in the direction of the gradient), then you are moving "as fast as possible" away from the level curve. That is what is meant by "maximum inclination"; if you think of $f$ as the height of a mountain, then the gradient is the direction which is locally the steepest.
The word locally is important. The gradient does not usually point to the top of the mountain. Once you leave the original point, the gradient may change direction and thus the locally steepest direction will change.