DIAGNOSIS This is usually the classic sign of a failed cooling system, most often a refrigerant leak or a compressor that's seized. The unit runs because the thermostat thinks it needs to cool, but the actual chilling mechanism is dead.
FIX First, unplug it. You'll need to check the condenser coils. On this model, they're the black grid of tubing on the back. Give them a good vacuuming—sometimes a thick layer of dust can cause it to overheat and shut down the compressor. After cleaning, plug it back in, set the thermostat to the coldest setting, and listen closely at the back. You should hear a distinct hum and feel vibration from the compressor kicking in. If it's silent or just clicks, the compressor is likely gone.
IF THAT DIDN'T WORK If the compressor runs but still no cold, the refrigerant has probably leaked out. There's no user-serviceable fix for that. It's a sealed system, and repairing it usually costs more than the fridge is worth.
HEADS UP Don't keep food in there while you're diagnosing. A fridge that runs but doesn't cool can actually spoil food faster than leaving it out, because it creates a warm, enclosed space.