Tile a region with tiny spinning cells. Every interior edge cancels against its neighbour, and only the boundary survives.
Every interior edge carries two arrows pointing opposite ways. They are the same integral with opposite signs.
Raise the grid resolution and watch the interior go quiet.
Green's theorem says that going once around the outside of a region gives the same answer as adding up the spin of every point inside it. Written out, that looks like a coincidence. It isn't — it's bookkeeping.
Chop the region into cells and walk around each one counterclockwise. Now look at any edge in the interior: it belongs to two cells, and those two cells walk it in opposite directions. The two contributions are the same number with opposite signs, so they cancel, exactly, no matter what the field is.
Every interior edge dies that way. The only edges nobody cancels are the ones on the outer boundary, which belong to just one cell. So the sum of all those little loops collapses to a single trip around the outside — which is why the two numbers in the corner agree at every resolution, not just in the limit.
The limit does one more thing. Shrink the cells and each little loop becomes curl F times the cell's area, which turns the sum into a double integral. Watch the third number climb toward the other two as you add cells.