This slide explains how, given a region on a plane, we can project it onto a sphere, then project that projection onto the plane of the sphere to get an area and solid angle.
We can find the projected area in this case by first projecting sector areas corresponding to each of the arcs in the curved triangle above. Then, we just sum the areas. The areas are signed based on the triangle winding order, so some areas will end up positive and some negative. N is the base plane normal and Ni is a sector normal for sector i.