During the chemical step of the jute silk manufacturing process, chemicals are added to the fibers to turn the jute into a nice, soft and comfortable yarn. Obviously not all jute fibers are treated in this process like burlap bags etc are quite rough when you get your hands on them.
The jute silk that is woven into clothing can be quite soft and pleasant to have next to your skin. Often the jute is covered with a second fabric to make it feel better next to your skin. The Indian saree is often made from a blend of jute or jute silk and this material is made to be nice and soft as well as more colorful.
