How Many Rectangles in the Grid?
Python, 22 bytes
lambda m,n:m*~m*n*~n/4

The formula m*n*(m+1)*(n+1)/4 is shortened using the bit-complement ~m=-(m+1), expressing (m+1)*(n+1) as ~m*~n.
Why is the number of rectangles m*n*(m+1)*(n+1)/4? Each rectangle is specified by the choice of two horizontal lines (top and bottom) and two vertical lines (left and right). There are m+1 horizontal lines, of which we choose a subset of two distinct ones. So the number of choices is choose(m+1,2), which is m*(m+1)/2. Multiplying by the n*(n+1)/2 choices for vertical lines gives the result.