If you assume that the sides of the original paper are of length a and b, with a greater than b, you can begin to work your way towards finding out the dimensions of the resulting rectangle by following along with the folds, and using the Pythagorean theorem (actually, you just need a very special case of Pythagoras - for triangles with 45 degree angles)
Thankfully, these simplify down to what I found to be a surprising result. The interior rectangle has an area equal to one quarter the area of the original rectangle. More surprising to me was that each side of the smaller rectangle depended only on one of the sides of the original - the longer side of the interior rectangle is 1/sqrt(2) of the shorter original side, and the shorter side of the interior rectangle is 1/(2sqrt(2)) of the longer original.
I'm sure there are probably easier ways to see this relationship. If you construct it in GSP or other dynamic geometry package you can experiment easily with increasing the side lengths.