What are Degenerate Rotational Energy Levels Physically?
I have two questions...
  
  Why does a molecule "gain potential energy" when it rotates? Does it
  want to stop rotating for some reason?
  
  What do degenerate energy levels correspond to physically in terms of
  the molecule's rotation?


Do you remember the Newton's law of motion? A body in motion will always remain in motion until and unless there is a force acting on it. So you can imagine, that once a force caused the molecule to rotate, the molecule will continue to rotate forever. As a crude analogy can you recall the time when the Earth stopped spinning on its axis? It is spinning for billions of years. In the same way you can think that the isolated molecule, once excited, will rotate for a pretty long time (on a spectroscopic time scale).

You can think of degenerate rotation as a a way of physical rotation of a molecule in such a way that if you calculate the rotational energy in each case, it turns out to be the same number. 

Third point, as pointed out by orthocresol, rotational energy is kinetic energy not potential energy. So no need to invoke potential energy.

Sometimes, we should not actually try to visualize what is happening physically at very small molecular levels. This is where mathematics enters. Many results originate out of pure mathematical calculations which can be hard to imagine mentally. For instance, Bohr's got correct H2 spectrum results by assuming the electron is rotating around the proton. The question is it actually rotating like a planet around a proton? A short story is worth sharing which will help you in future. Feynman (Physics Nobel) used to think a lot about of photons as a student and discussed them with his father. When he finished his PhD in physics, he realized, he still could not explain what a photon is. Sorry I cannot recall the reference. But mathematics has no problem. Yet on the web we can "see" photons in pictures as balls, which are obviously pure but meaningless imaginations.