AgCl is soluble in ammonia because it form complex [Ag(NH3)2]Cl. The formation of the complex [Ag(NH3)2]+ ion makes it behave like ionic therefore making it soluble. If we add nitric acid it will break the coordinate bond between silver and ammonia molecule, the ammonia will react with nitric acid to form ammonium nitrate and the white precipitate of silver chloride will reappear.
AgCl+2NH3→[Ag(NH3)2]Cl