The rings have formed where they are because there is greater gravity there. The reason is the shape of the gas giant. It is far from circular. It is wider at its equator. Centripetal force causes a thicker equator. Therefore because the thickness is far wider at the equator, the gravitation at the equator is more than that at the poles. There is more mass pulling you down at that point. The particles of the rings fall into a gravity well.
This explanation is self evident because as can be seen from the above photo, the rings are so incredibly thin, and are PERFECTLY aligned with the equator.
Compare the reason why the moon only shows us one face. It must have more mass on one side. It is like a ship floating on the ocean. The ship has more mass in its hull that gravity pulls into the water.
On the gas giants, given time every asteroid has worked their way to the equator.
I would imagine being an asteroid, it would take a very long time of maybe thousands of years and collisions to achieve such a thin presence. It is very beautiful though!