Why do they cut the fibers after braces?

Fibers?: Not sure what you mean by fibers. If you're talking about a circumferential fiberotomy, that's done pretty infrequently but is done to separate the periodontal ligaments that connect one individual tooth's roots to its corresponding socket so that when the periodontal ligaments re-attach to that particular tooth, they will re-attach to the new position that the tooth has now been moved to.

Braces: If teeth were severely rotated or had to move a lot then those fibers connected to the teeth will want to pull the teeth back to where they were before your braces. Cutting the fibers can give you better stability.

Stability, but no: Fiberotomies are done to increase the stability of the orthodontic correction. The fibers that are cut are above the bone, and for that reason i doubt its effectiveness. The overwhelming majority of fibers that connect the root to bone are below the crest of bone, different vertical levels have different angulations. If only 5 or so % of fibers are cut, there are still many more to relapse.