This term may mean different things to different weavers. For me, it means putting them in a small bucket of water overnight.
On the loom After wet finishing
A term long-used in the weaving business is fulling, most often used when referring to woolen cloth. By wet finishing, the fibers are encouraged to shrink up and close the gaps between individual threads, making the fabric fuller. If you look closely at the two photos above you will see that the band has shrunken up a bit in the water, and the pickup pattern looks nicer because of it. The individual threads are not so much seen as individuals now, but are closer together, giving a more graceful line to the pattern.
Another thing that soaking or washing does is allows the colors to bleed if they are going to. I have often been surprised by how much color comes out of the yarn into the water. To keep this color from being absorbed back into the band, I use the miraculous Shout ColorCatcher sheets. I buy them in the laundry section of the local supermarket and they do exactly as their name implies. Most of the time I find that half of a sheet works fine for one woven band. If a lot of color is bleeding out, I change the water a few times but leave the sheet in. I've often wondered how I could use these interesting dyed sheets in some way. Please let me know if you have ideas!!