Every smooth surface is flat if you look closely enough. That flat thing is the tangent plane.
Zoom in and the gap collapses, but the gap ÷ h² settles on a constant. That constant is the curvature you're leaving out.
Zoom in and watch the gap between surface and plane collapse.
Differentiable means one thing: locally flat. Zoom in far enough on any point of a smooth surface and it becomes indistinguishable from a plane — and that plane is the tangent plane. Everything else about derivatives in several variables is bookkeeping around this one fact.
The two slices show where the plane comes from. Hold y fixed and you get a curve whose slope is fx; hold x fixed and you get another with slope fy. Those two tangent lines cross at the point, and a plane is exactly what you get by spanning them. That's why two partial derivatives suffice.
Watch the middle readouts rather than the picture, though, because they say something the picture can't. The gap between surface and plane goes to zero — but so would a bad approximation. What makes this one the tangent plane is that the gap dies faster than the patch shrinks: divide it by h and it still goes to zero. Divide by h² and it settles on a constant instead, which is the precise sense in which the error is second order.
That leftover constant is curvature, and it is exactly what the second-order term captures. Turn the quadratic on and its own gap dies like h³ instead — one order better, which is what buying one more term in the Taylor series gets you.