It's not a stain -- a chemical discoloration -- at all. What it is is solid, powdery material, iron oxide, that is lodged in the nylon fibers. So, there are two approaches you can take. You can chemically react the Fe2O3 and change it into something else which doesn't have that annoying red color, or you can mechanically remove those tiny bits of iron oxide.
The commercially-available product to accomplish the first is called CLR, calcium, lime, and rust remover. You can get it at most grocery stores.
But, I'd start with the first approach. Try vacuuming; suck those little bits of rust out. Try running compressed air through the nylon from the other side to blow those bits out. Try running pressurized water through from the non-"stained" side to flush them out. Try a foaming cleaner that will lift them out.