Soft armor is made from Aramid materials, the most common of which is Kevlar. Kevlar is a very strong, and heat-resistant synthetic fiber, and its individual strands can be five times stronger than steel on an equal weight basis.
These kevlar fibers are tightly woven to make kevlar sheets, which are stacked together to make soft armor inserts. When a fired bullet makes contact with a bullet-proof vest, it can’t break through the tightly woven fibers and tries to push them out of the way to make a path through them. This absorbs the energy of the rounds, and dissipates it on a much larger area, catching the bullet in the vest as well.