Let’s look at an example of this.  Suppose we have an image in which H(x,y) = y.  That is, the image will look like:
11111111111111
22222222222222
33333333333333
And suppose there is optical flow of (1,1).  The new image will look like:
-----------------------
-1111111111111
-2222222222222
I(3,3) = 2.  H(3,3) = 3.  So It(3,3) = -1.  GRAD I(3,3) = (0,1).  So our constraint equation will be: 0 = -1 + <(0,1), (u,v)>, which is 1 = v.  We recover the v component of the optical flow, but not the u component.  This is the aperture problem.