For the very sensible reason that they’re hungry and they like the taste of wool. The eaters are actually the larvae, or caterpillar stage the moths go through before they get their wings. The caterpillars have mouths with strong, biting jaws that can chew cloth and fur.
When the caterpillar changes into a winged creature, a moth, its mouth changes too, and it can only suck nectar from flowers through a hollow tube at the front of its head.
You may wonder how these tiny crawling caterpillars get to the wool clothing in the first place, since they cannot fly. This is because the female clothes moth lays her eggs on clothing and furs and in rugs and carpets. When the larvae hatch, they first eat their eggshell, then begin to eat whatever material they rest on.
While caterpillars have very poor eyesight and can see only light and dark, adult moths have keen vision through the thousands of eyes on each side of their head!