The earth is constantly radiating heat in the form of infrared light. During the day, sunlight generally warms the earth faster than the earth looses heat. At night, temperatures drop as heat radiates away with nothing warming the earth up. Clouds can act like a blanket by reflecting some of that heat back to the surface. This keeps the temperatures from getting quite as cold.
On clear nights, the earth still radiates heat as infrared light. However, with no clouds to reflect it back the heat is lost into space. When combined with already cold conditions, this can make for very cold nights.
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