In your first link the narrator says he "doesn’t understand the physics of it" but there's really no physics involved (ignoring scatter). It’s just a consequence of the math. It’s relatively easy to understand if you think of it in terms of the surface of a sphere. There is a fixed amount of light coming from a point source, and as the light travels outward you can think of it as being spread over the surface of a sphere. Since the surface area of a sphere is 4pir^2, if you double the radius the area quadruples, and therefore the light intensity at any point on the sphere drops by a factor of four.

edit: And now after rtfm I see there's a nice demo of this!