Refrigerators rely on the phase change of specially formulated refrigerant from a liquid to a gas. The compressor pressurizes and super-heats a gas, which, when pushed out the back of the refrigerator, releases a bunch of heat into the air. This temperature drop at such high pressure causes the gas to liquefy. This liquid boils at a very low temperature, so as it enters the refrigerator it immediately absorbs "heat" and boils into a gas on its way back to the compressor. This boiled gas, which "heated" as it went through the refrigerator, is what makes everything else cold. The air that touches these coils become cold as they heat the refrigerant.

I thought that would come out shorter. I hope it makes sense to you. The important thing to understand is that the refrigerant (a dangerous, regulated chemical) boils at a temperature well below freezing, but by compressing it with a pump, you're able to achieve the pressure needed to turn it into a liquid, which pretty much immediately boils.