Bend a grid and watch a tiny square stretch. The area it gains is exactly the factor the integral needs.
Shrink the square and watch the measured ratio settle onto |J|. That limit is the whole definition.
Drag the little square around and shrink it.
A change of variables is a way of redrawing a region so its description gets easier — a disk becomes a rectangle in polar coordinates, and rectangles are what integration limits like best.
But you cannot just swap the variables and carry on, because the transformation distorts area. The uv grid on the left is made of identical squares. Its image on the right is not: near the origin in polar coordinates the cells are tiny, and far out they are large. Any honest total has to weight each cell by how much area it actually occupies.
That weight is the Jacobian. Zoom in far enough and any smooth map looks linear, a linear map multiplies every area by the determinant of its matrix, and that matrix is the one built from the four partial derivatives. So |J| is the local area-stretching factor — the number that turns du dv into dx dy.
The two numbers in the corner are the claim and the evidence: |J| computed from derivatives, and the honest measured area of the image quadrilateral divided by the area of the square that produced it. They disagree while the square is big, because curvature matters at that scale. Shrink it and they converge — which is exactly what "infinitesimal" is doing in the formula.