There is significantly less energy available to heat the soil. This has the effect of reducing the amplitude of the daily temperature wave. Clouds affect the diurnal temperature range as well. Clouds more than a few hundred meters thick reflect sunlight efficiently, about as well as new fallen snow, thereby reducing the solar energy reaching the ground. This lowers the maximum temperature that would be attained if the sky were clear.
Warmer ground radiates significantly more energy than colder ground.
Clouds absorb infrared radiation very effectively and re-radiate it back toward the ground at a rate dependent upon the cloud temperature. The temperature of low clouds is likely to be close to the temperature of the ground. Even when the sky is clear, water vapor reduces the diurnal temperature range. The biggest effect of atmospheric moisture when the sky is clear is to elevate the overnight minimum temperature because the water vapor in the atmosphere absorbs some of the infrared radiation from the ground and re-radiates a portion of the absorbed energy back toward the ground. In a dry atmosphere, more infrared radiation would pass to space, and thus surface cooling would be more pronounced.