Bend a flat sheet of graph paper into a shape. Where the sheet stretches, every square on it has to be counted for more.
Drag the patch on the flat sheet; drag the surface to turn it.
A surface is a sheet of rubber. The rectangle on the left is the sheet before you bend it, ruled into identical squares; the shape on the right is where each of those squares ends up. Every point of the surface has an address (u, v) back on the flat rectangle.
Hold v fixed and let u run: you trace one of the curves drawn on the surface, and its velocity is ru. Hold u fixed instead and you get rv. Those two vectors are tangent to the surface, so together they span the tangent plane — and their cross product is perpendicular to it. That's where the normal comes from; you never have to guess it.
The cross product's length is the other half of the story. A tiny rectangle du by dv on the flat sheet lands as a tiny parallelogram with sides ru du and rv dv, and the area of a parallelogram is the length of the cross product of its sides. So dS = |ru × rv| du dv — the sheet stretches by exactly that factor, and any integral over the surface has to pay it.
The two numbers in the panel are that claim, checked. One is |ru × rv| at the patch's centre; the other is the patch's honest area divided by the du dv it came from. Shrink the patch and they meet. It is the same argument as the Jacobian, one dimension up — and on the sphere it is why the coordinate squares near the poles collapse to slivers while the ones at the equator stay fat.