Therefore, in the longwave, clouds work to warm the Earth. This is the opposite of the effect of clouds in the shortwave. The main difference between high and low clouds is that the high clouds are colder. This means that the emission of radiation to space by the high clouds is less than for lower, warmer clouds. So there you have it.
High clouds are poor shortwave reflectors but efficient longwave blockers.
Thus, high clouds will cause a warming of the Earth's surface. Low clouds are excellent reflectors, but less efficient at keeping the longwave radiation from escaping to space. Therefore, low clouds help to cool the Earth. There is an excellent NASA fact sheet on the effect of clouds on the Earth's Energy cycle. Once we have that, it's fairly easy to match the two pieces of data.