Silk is a natural protein fibre, and the agents chosen to finish it are usually ones that stay chemically compatible with that protein structure so they bond well without damaging the delicate strands.
The stiffening agent traditionally used to finish silk fabric is gelatin. Gelatin is itself a protein obtained from animal collagen, which is why it blends so naturally with silk's own protein-based fibre. When dissolved in warm water and applied to the cloth, it settles around the individual fibres and forms a very thin protein film as it dries.
This film gives the fabric a firmer body and a crisp handle without turning it papery or brittle, and being water-soluble and biodegradable, it washes out cleanly later without dulling silk's natural sheen. This is precisely why gelatin, rather than a mineral or starch-based agent, is the finish associated specifically with silk among the different textile fibres.
Therefore, the stiffening agent used for finishing silk fabrics is gelatin.