What movements happen at this joint?
Rotation happens at the trunk when you turn your trunk to one side or the other in the transverse plane.
What muscles make these movements happen?

Your external and internal obliques rotate your trunk. Contracting either external oblique produces contralateral rotation—i.e., contracting your right external oblique rotates your trunk to the left. Contracting either internal oblique produces ipsilateral rotation—i.e., contracting your right internal oblique rotates your trunk to the right.
What exercises make these muscles work?
Twisting ab exercises like bicycles and russian twists target the obliques.