Fulling is a step in clothmaking which involves the cleansing of cloth (paricularly wool) to get rid of oils, dirt, and other impurities. Its name comes from the fact that the cleaned cloth is left more "full" (springy of texture) than was the dirty cloth before fulling.
A fuller is the worker who does the job. In the process, fuller's earth was used. (This is a soft earthy material occurring in nature as an impure hydrous aluminium silicate.) The cloth might also be washed. After washing, to prevent shrinkage and wrinkling, the cloth would be stretched on great frames known as tenters and held onto those frames by tenterhooks. It is from this process that we derive the phrase being on tenterhooks as meaning to be held in suspense.