I think that all that is wrong is a burned-out connection to the relay.
The large yellow and black item is a transformer that produces low-voltage electricity to run the electronics. The electronics control when the fridge compressor runs. The blue relay has a low-voltage coil, controlled by the electronics, and the relay is what actually switches the compressor on and off.
The tracks on the board go from the L connection, through the relay, and to the Comp connection. These tracks have to take more current than anything else on the board, and if there is a bad connection, there will be a lot of heat.
The weight of the relay along with vibration has probably stressed the connection, and then the heat from the damaged connection has caused it to burn out.
You probably just need to
1) clean off the black marking, which will conduct electricity, so could cause a short.
2) solder wires in place of the burned-out track.
It's possible that the fuse has blown which will need unsoldering and replacing. Test the continuity to see if it has blown.
The relay could be damaged, either if the pin is burned off, or it could have been damaged as the heat built up, or it could have been the cause of the heat that initially weakened the solder joint. Relays like that aren't repairable, but they are easy enough to find replacements for. Unsoldering them is a bit tricky as you need to get solder off all of the pins.