Not to belabor the point, but hopefully to help really grasp the concept:
To really visualize the difference between shifting and simply raising/lowering the camera, I like to think of the shift function as in-camera cropping of a wider FoV image.
For example, let's say that you want to capture a building in the frame and the FoV of the lens that you're using gives you *just* enough width to fit the entirety of the building in to the frame *if* you point the lens up. Unfortunately, pointing the lens up will result in that perspective distortion - the "falling in" or convergence of upright parallel sides. To avoid this, you could go to a much wider lens, compose in a way that keeps the lens pointing straight (perpendicular to the building, not up, thus avoiding the falling in), take your capture and then crop out the unwanted stuff in post. The problem with this approach, of course, is that you're throwing away a lot of pixels that aren't on-target and you're getting a much lower resolution subject.
A shift, or perspective control, lens effectively does the above, without throwing away the pixels, by giving you a much wider FoV (big image circle) and then letting you crop it as you please before hitting the shutter.
