The cooling is inherent in the design, hence why they are close together.

 

The heat of the stove is kept away from the refrigeration unit with only the pipes necessary to carry the hum from engine to the fridge connecting the two.

 

The thermo-acoustic engine is essentially a Stirling engine with no moving parts. (other than the gas). Burning wood heats the pipe at one place and cools it at another.  The pipe then resonates, rather like an organ pipe. The gas moves from the hot part, and expands, to the cold part and contracts. Just like a stirling engine this transfers energy to the gas and makes the hum louder. To generate the cooling effect, this hum is passed to another thermo-acoustic engine working backwards. The hum causes part of the pipe to heat up and another part to cool down. The hot part is kept at ambient temperature and so the cool part becomes a refrigerator.