DIAGNOSIS: That's the ice maker's ejector gear or auger motor failing. The plastic gears are a common weak point and the noise means they're trying and failing to turn.
FIX: You need to replace the ice maker assembly. Unplug the fridge, remove the ice bucket, and look for two screws securing the ice maker. Remove them, disconnect the wiring harness plug, and install the new ice maker. Plug the fridge back in.
IF THAT DIDN'T WORK: The noise could be from the compressor's start relay failing, which also produces a rapid clicking. That requires a technician to diagnose and replace the relay.
HEADS UP: Don't ignore this. Letting the broken gears grind can cause them to seize completely and potentially burn out the motor, leading to a more expensive repair.