The easiest method is to brush or hand remove those little bits of fiber each day as you see them. Otherwise, they tend to accumulate more fiber, which holds in the manure and soaks up the urine, which sticks to more fiber. . .it takes very little time for that tiny molting clump of fiber to become a thick mat surrounding the wire on both sides, which is difficult to remove.
I have a long-handled car scraper/brush that allows me to reach inside and beneath the cage, brushing out any loose fiber or dangling manure that stuck to the fiber, before it becomes a problem. I have a flat, small backpacking shovel that fits inside the cage for removal of any sodden mats that were missed and don't brush, as well as a scrub-brush on a long handle and a short handled one.
My mentor used the torch on her cages, but it seems like that damages the finish on the wire. Her cages didn't last that long before needing replaced, and I've got some that are 30 years plus and still in good condition. The secret seems to be in not allowing any manure/urine build-up, which rots out the wire quite quickly, especially with smaller gauge floor wire. (14 gauge galvanized-after-weld GAW wire is standard, 16 gauge is thinner and more likely to bow and rust but is used by some of the pre-built cage companies, 12 gauge sometimes is recommended for giant breeds.)
I don't have running water at the rabbit barn, so I don't use a power washer to remove hair, and I don't want to damage the galvanized finish, so I don't use a torch. So, I hand remove clumps as I see them, and use the brush as needed. I have one doe that seems to continually molt fiber that sticks to the wire terribly. It is a daily brushing challenge, and not one I want to encourage in the herd.