Gasses heat up when they're compressed (squeezed) and cool down when they expand. If you push gas around in a loop, and you have half the loop set up so that it compresses the gas, and in the other half you expand it, you'll have a loop with a hot side and a cold side.

Refrigerators work by using a loop like the one I described above. They put the cold side in the box and the hot side outside the box. Because the cold side is colder than the food and air inside the box, it absorbs some heat. That heat moves with the gas to the hot side, and because the hot side is hotter than the surrounding air it expels some of the heat it's gathered, cooling the gasses again.

Continuously moving heat from inside the box to outside the box means it stays cold and keeps your food chilled.