The primary mechanism at work here is the type of clouds that exist at different altitudes.
Low clouds of the cumulus or strato-cumulus variety are optically thick and reflective. These types of clouds will scatter more light than they transmit or absorb, and the energy scattered upward and back into space will not contribute to warming.
High clouds outside of convection tend to be cirrus clouds, which are icy and not optically thick. These clouds let light pass through freely (you can see the sun through these clouds). This lets solar energy through but they will absorb longwave (terrestrial) radiation, and this is what will contribute to warming.
This is where a lot of the uncertainty in cloud feedbacks in a warming climate comes from. If we get more clouds -- low, bright clouds, then this could contribute to cooling, wheras high, transparent (to shortwave) clouds would contribute to warming. Again, it is not so much the high vs low that matters here, it is the properties of typical clouds at these heights.