What is the reason for the water spilling when the tray is rotated at high speed?
 
The water  of the tumbler when the tray turns around fastly, because initially water is filled in tumbler placed on the tray was in  now when we are turning we are actually applying a force on tray .The tumbler which is in contact with the tray  with the tray but the water placed in the tumbler  due to inertia of rest, hence the water  of the tumbler.