Why is the molar solubility of silver chloride lower in an aqueous solution of silver nitrate than in pure water?

Answer is: because silver cations are common ions in both chemical dissociation in water for this salts, so silver cations shift dissociation od silver chloride to the ledt according Le Chatelier's principle.

Chemical reaction (dissociation) of silver chloride in water:

AgCl (s) → Ag⁺ (aq) + NO₃⁻ (aq).

Chemical reaction (dissociation) pf silver nitrate in water:

AgNO₃ → Ag⁺ (aq) + NO₃⁻ (aq).