Leather belts do shrink when they get wet. When a belt gets wet, the water mixes with the leather's natural oils and removes those oils when the water evaporates, shrinking the belt and drying it out and making it more brittle.
How much the belt will shrink depends on the quality of the leather. Top-grain leather tends to shrink less since it’s made from the top half of the hide and is more resistant to shrinking, but genuine leather (a step down from top-grain) comes from the bottom half more prone to shrinking.