Focus breathing is generally a consequence of the lens’s focus mechanism, not its sole purpose. Changing focus shifts the optical relationships inside the lens, especially the position of the rear principal point relative to the image plane, and that can alter the effective angle of view.

This is not unique to internal-focusing lenses. Different focusing methods can produce breathing, including designs that move rear elements. Whether the field of view increases or decreases with closer focus depends on the optical design. In broad terms, telephoto and retrofocus designs can behave differently.

So the best summary is: lenses usually breathe because they focus, not breathe in order to focus. Internal focusing is one way to achieve focus, and focus breathing can be a side effect of the optical compromises involved.