You can breathe through dry cloth because the fibers in the cloth are woven together in a way that leaves tiny gaps between them that air can pass through.
Water plugs these gaps, and therefore makes it very difficult for any air to pass through said gaps.
Wet cloth blocks the tiny air gaps that let air through. Water fills the holes, so air has to push harder to get past, making it much harder to breathe.
It’s not so much impossible, per se, as it is inhaling water droplets. The wet cloth holds that liquid and when pressure, from breathing, pulls on it that liquid then goes into the lungs
A cloth is already restricting airflow because of all those fibers in the way, but there are enough gaps to let some air through. Damp or wet cloth now have a bunch of water droplets that are filling those small gaps in between the fibers, which further restricts airflow through the cloth.
And at that scale, water is really sticky. It loves to stick to other water, and it also loves to stick to some types of fabric.