I'm not sure I like the phrase "sums of 2D planes"- you can't add planes, they aren't numbers! You can think of as the volume under the surface z= f(x, y) and the rectangle in the xy-plane, a< x< b, c< y< d. Using Riemann sums, we divide that region into tall thin rectangular solids, having base and height where is some point inside one of the small rectangles. Yes, because, at each step, we have only a finite number of such rectangles, it does not matter whether we add the volume "horizontally first, then vertically" or "vertically first, then horizontally". That is the "meat" or Fubini's theorem. All the "hard" work is hidden in the limit process! And we know that works precisely because the function is "integrable".