DIAGNOSIS The compressor is running but the refrigerant isn't circulating. Most likely cause is a failed start relay on the compressor.
FIX Unplug the fridge. Locate the compressor at the back bottom. The start relay is a small black or gray plastic box clipped onto the compressor terminals. Unclip it. Buy a universal 3-in-1 hard start relay (compatible with small compressors). Plug the new relay onto the compressor terminals. Plug the fridge back in.
IF THAT DIDN'T WORK The compressor itself may have seized. You'll hear a click then a hum that stops. This is not user-serviceable. The unit needs professional repair or replacement.
HEADS UP Do not bypass the relay or try to start the compressor manually. You can damage the control board and create a fire hazard.