How do I repair this leaking toilet supply nut?
There will indeed be some type of rubber, silicone or other type of flexible materal washer thing inside the nut. The cross section picture below attempts to show how the typical compression fitting of this type works.



As the nut tightens it compresses the "rubber" part against the end of the threaded plastic piece and around the outside of the incoming water line. Leaks can arise as from this "rubber" part getting hard and inflexible. The chorine added to water can accelerate this this problem. Over tightening the compression nut will also lead to problems. Sometimes a corrosive build up on the water tube can also lead to leak issues. Usually it is combination of all three elements and everything is OK till there is some flexure in the joint and then it starts to leak.

To fix it is suggested to replace the compression nut and the "rubber" part. If the inlet tubing is corroded then clean it up with steel wool. If the plastic threaded part is stripped from previous over tightening then you will also have to replace the fill valve assembly in the tank too. You should be able to find replacement parts at hardware or home stores. Replacement parts are usually fairly reasonable in cost.