Silk cloth is made from silk thread. Silk thread can be gathered from the webs of brown recluse spiders, brown recluse spider men, cave spiders, giant cave spiders, and phantom spiders (as well as any web-spinning forgotten beasts, titans, or clowns), which is then woven into cloth at a loom. Webs are automatically collected by any worker who has the weaving labor activated and access to a loom, which automatically produces the work task "collect webs" and turns collected webs directly into silk thread. No stockpile is necessary for this collection.
Automatic collection of webs can be disabled from the orders menu (o -> W) - this is useful if the accessible webs are located in a dangerous area or if you want to temporarily free up your weavers for other tasks.
Silk is generally a superior material for clothing as it is a lighter material than leather and far more protective than other cloth materials in the event of animal attacks on civilians.
 See also