Paper is made up of a sheet of short fibrous strands, which when dry are hard and hold their shape. With a traditional fold, the fibers in the paper break in half creating what is commonly called a crease. This crease can be used as a hinge or as a perforation that can be teared. The more a point or line is folded, the more fibers break, and the weaker it becomes. One side-effect of a design with a lot of folding is that it often tears accidentally.
When the paper is wet, the fibers soften and become much more flexible. A properly moistened sheet, when folded, will not break the fibers, but bend them. Since the fibers are undamaged, there is no crease, only a fold. This makes the folded points and lines much stronger throughout the process and when the model is complete. It also means that shapes can be molded onto the paper while wet, and once dried it will hold the new shape.
What I have learned of wet folding comes mostly from a book by Michael G. LaFosse, who founded Origamido Studio. He creates his own paper and usually employs various wet folding techniques.