Why does a single phase synchronous motor need a capacitor but for a three phase motor it is not required?

A three-phase motor has a rotating magnetic field inside that is set up by the supply, so the armature always rotates in the correct direction. A single-phase motor can in principle start either way so it has a starting winding supplied through a capacitor to provide a small component of rotating field to get it started the right way. After it has picked up speed the starting winding can be switched out.