There’s radiative cooling from all altitudes up to the tropopause (stratosphere higher up is of lesser importance and very different). With more CO2 less and less radiation can escape directly to the space from the surface and lowest troposphere and a larger share originates at high altitudes. One part of this change is the increase in the altitude of tropopause. All these changes contribute and work in the same direction.
Pushing tropopause higher is important, because the lapse rate remains high and essentially constant throughout the troposphere. Higher tropopause means therefore that the temperature difference between the surface and the tropopause is larger. Therefore the Earth looks colder from the space and radiates less to the space until the temperature of the Earth has risen enough to compensate that effect.
Assuming that CO2 is added over a short period, the rapid response is reducing the radiation from earth to space. That leads to warming and the final response is warmer Earth surface. At that point the effective temperature of Earth as seen from the space has returned back to the earlier level, but the new effective temperature is a combination of colder tropopause and warmer Earth surface (and also warmer lower troposphere at fixed altitude).