The fans are running, so it sounds like your refrigerator is okay. But the compressor—which constricts refrigerant to cool it—is doing nothing, even though that clicking sound is supposed to be the start relay turning the compressor on and off depending on the thermostat reading.
The problem is that the start relay isn't triggering the compressor into action. So, whatever else you can see and hear, there's simply no refrigerant being cooled and circulated.
Depending on your appliance’s make and model, you may be able to remedy this issue yourself by installing a replacement part:
- Unplug your refrigerator from the power supply.
- Move the fridge away from the wall so you can access the rear panel.
- Locate the compressor; the start relay will be mounted to the side.
- You should be able to remove the start relay by hand.
- Shake the relay gently; if it rattles, it is almost certainly faulty.