Let's for a moment think about the troposphere consisting of stacked layers. The anvil clouds can only be found in the upper layer of the troposphere, since this is where they have to extend horizontally due to the tropopause acting as a natural barrier (see Figure 1. in Hartmann, Larson linked below). Let's assume that moisture does not penetrate the tropopause or stratosphere and that layers containing moisture fully absorb long wave radiation (this is obviously not true but it helps here to get to the point). This means that the outgoing long wave radiation from this upper layer can not be absorbed by water vapor by layers above them.
Since the layer containing the anvil clouds will absorb all outgoing long wave radiation from the layers below (by our hypothesis stated above), we can conclude that this layer is the single most important layer that determines how much radiation is going to space and thus, cools the atmosphere.
This explains how an atmospheric layer can contribute to cooling but does not explain why cooling decreases with decreasing water vapor mixing ratio. The answer can be explained in terms of the atmospheric lapse rate.
The dry adiabatic lapse rate in the atmosphere is about $\Gamma_d = 10$ K/km. The moist adiabatic lapse rate is about $\Gamma_w = 6$ K/km. The more the water vapor mixing ratio decreases, the closer the lapse rate will be to $\Gamma_d$. And thus the upper most layer of the troposphere will be colder compared to a situation where the water vapor mixing ratio is high. A colder top layer will radiate less and therefore cooling decreases, since there is less outgoing long wave radiation.
The FAT assumes that the temperature of this upper atmospheric layer is independent of surface temperature and thus, would amplify warming. What changes is the height of the upper layer - the troposphere has a larger vertical extent in a warming climate.
Hartmann, Larson - An important constraint on tropical cloud - climate feedback