Point a compass needle anywhere on a contour map and watch the slope in that direction respond. One direction always wins.
The needle is a unit vector. Turning it changes only the direction you step, never how far.
Drag the dot to move, drag the needle to aim.
The directional derivative asks a simple question: if I take one small step in direction u, how fast does the height change? Turn the needle and read the answer off the dial.
The dial is the whole story. Plot that rate in every direction at once and the answer traces a circle — because the rate is a dot product, and a dot product with a fixed vector is a cosine. The circle's far point is the direction of steepest ascent; the two directions where the circle passes through the centre are the ones where nothing changes at all, and those are exactly the directions along the level curve.
So the gradient is not just a useful direction. It is the diameter of that circle, it always points perpendicular to the level curve, and its length is the steepest rate available. Everywhere except a flat spot, where the circle collapses to a point and every direction is equally pointless.