Flip your refrigerator inside out. That is exactly what is happening.

You need to understand the difference between "heat" and "temperature".

When you squeeze a gas inside a perfectly insulated container, no heat is transferred, but the temperature increases. If you allow the gas to expand again, it will return exactly to the temperature it started at. Again, no heat has transferred into or out of the gas, due to our (hypothetical) perfect insulation. The temperature has changed, but no heat has moved.

In reality, we aren't using a perfect insulator. When we compress the gas, the temperature of the gas rises. Heat starts flowing from the compressed gas to the container, and into the air. The gas is still compressed after the heat has left, but its temperature falls to ambient. This all happens inside the house.

The cooled, compressed gas is now piped outside. It is expanded. Because it doesn't have the same amount of heat that it started with, the temperature plummets. It gets extremely cold, far colder than the outside atmosphere. Heat from the atmosphere flows into the cold gas. The cold, decompressed gas then goes to a compressor, and the cycle repeats.