Can someone please explain to me in simple terms how heat exchangers might be used to cool food?
Cool relative to what? - hot food cooled to room temperature could involve collecting the heat from the surface it is laid on by using liquid or by passing air over it and taking the warmed liquid/air off to a heat exchanger to have the hear removed without exposing the food to the passing dusty/dirty/germ-laden air. Cooling food below room temperature requires a bucket of ice or some kind of refrigeration cycle which almost automatically includes one heat exchanger in the cycle and could include a second used as above.
It's a refridgerator you're looking for: The inside of the fridge absorbs heat from the food and passes to the outside of the fridge via the heat exchanger