Why Cloudy nights are warmer than the clear nights?

During the day, the earth receives light from the Sun and gets heated. Now, during clear nights (when there are no clouds), the heat from the earth surface is transferred back into the space through the atmosphere resulting in cooling of the earth. Now, as when the clouds are present during night, the heat can't be able to escape. The clouds acts as a barrier. Since, the heat remains present in the atmosphere, the cloudy nights are warmer than the clear nights.