The essential principle behind the lapse rate is that as pressure decreases a given parcel of air will expand, in the process having the ability to do work, therefore losing energy and as a result cooling. Pressure decreases roughly as an exponential function of altitude.
The lapse rate plays an essential role in insuring that the upper layers of atmosphere are cooler and thus that as the levels of carbon dioxide in the atmosphere increase the altitude at which radiation that escapes to space gets emitted is cooler, resulting temporarily in an imbalance between energy entering the climate system an energy escaping the climate system until the upper layers warm enough so that the temperature where radiation is emitted that escapes to space is nearly the same temperature as the lower layer prior to the rise in the level of carbon dioxide. But in the process, given a roughly constant lapse rate, the temperature of the surface must rise.
