Fulling is a process of beating woollen cloth to soften it and felt the threads together so that cut edges don't fray. The clay is used to soak up the wool-grease (lanolin).
It was demonstrated recently on telly by the excellent Tony Robinson, as one of his "Medieval Occupations Which Not Even Baldric Would Do" or whatever it's called. You also need rancid urine to remove the grease.
Nowadays fuller's earth is used as drilling mud (that is, lubricant for oil wells etc), and as Andy says for absorbing oils -- but I think the largest use now is for cat-litter.
The phrase Andy quotes (which I must say I've never heard) must just be a pun on the term.
Note that "fuller" in this case is not someone's name, so it doesn't have to have a capital initial letter.