I can explain it for rings.
The phenomenon you're looking for is called nodal precession. Meaning, when the planet has an equatorial bulge, all non-equatorial orbits will precess, will wobble around the spin axis.
Because of the bulge the gravitational force on the satellite is not
directly toward the center of the central body, but is offset toward
the equator. Whichever hemisphere the satellite is in it is
preferentially pulled slightly toward the equator. This creates a
torque on the orbit. This torque does not reduce the inclination;
rather, it causes a torque-induced gyroscopic precession, which causes
the orbital nodes to drift with time.
For rings, any such precessing orbits would be unstable - ring particles would collide with each other. The only non-precessing, non-colliding orbits are the equatorial ones. So that's where the rings persist.
As for satellites, I would venture a guess that their orbits actually do not migrate towards the equatorial plane, they stay where they are, just slowly precessing. Someone please correct me if your orbital mechanics is in better shape than mine.
EDIT: There's a ton of perturbations from an oblate planet, and some do apply to the inclination of the orbit. So it's not just precession. Some terms are periodic. Google "inclination of orbit oblate" without the quotes. It's... complex. I'll edit this answer tomorrow if I find something relevant.
EDIT2: This is far more complex than I thought. A thorough treatment is given in Kozai, Y. The motion of a close earth satellite:
Chapter 11.15 in J.M.A. Danby, Fundamentals Of Celestial Mechanics, is entirely dedicated to this topic. He talks about critical values for initial parameters where his results are not applicable, so obviously there isn't a simple conclusion that applies in all cases.
I am not going to interpret the math, sorry. That's all I had to say on this topic.