The water spills out of the tumbler when the tray turns around fastly, because initially water is filled in tumbler placed on the tray was in state of rest now when we are turning we are actually applying a force on tray .The tumbler which is in contact with the tray also starts moving along with the tray but the water placed in the tumbler remains in rest due to inertia of rest ,hence the water spills out of the tumbler.