What is the cloth around my dishwasher and do I need it?
It's called an insulation blanket and its main purpose is to muffle the noise made during operation. Since it's usually made of fiberglass, it provides some heat insulation too.

You don't need it, but you'll probably find the extra noise to be annoying. Try the dishwasher without it, and if you can't live with the noise, replace it.