GHGs make the atmosphere opaque to certain bands of IR. let's concentrate on CO2, and ignore convection which is primarily an effect of water vapour.
roughly 8% of the surface's power is emitted in the 15 micron band dominated by CO2, and it is absorbed to extinction by 10 meters of atmosphere. the exact numbers dont matter, just the general mechanism. so, all that energy is added to the first 10 meters of atmosphere but what happens to it?
when a molecule absorbs a photon it increases its potential energy, but we know that potential and kinetic energy of the atmosphere is freely transferable by collisions. therefore adding potential energy will result in increased kinetic energy (temperature) as well.
the equipartition theorum states that for a substance at a consistent temperature the emission will equal the absorption, basically a restatement of emissivity. this is where the warmers make their claim that CO2 absorbs, then emits in a random direction, which means roughly half returns to the surface. this is incorrect because the atmosphere cools with height because of gravity and the distribution of potential and kinetic energies. (I am ignoring water vapour contribution to lapse rate per above)
every slab of the atmosphere contains some excited CO2 (potential energy), from radiation coming both from above and below. it continues to hold onto this energy until the concentration of CO2 is so thin that emitted photons are more likely to escape to space rather than encounter another CO2 molecule and be absorbed. the amount of CO2 specific radiation lost to space is a small fraction of the amount of CO2 specific radiation put into the atmosphere by the surface. the difference is used to add to the total energy of the atmosphere which both warms the air (kinetic) and increases the height of the atmosphere (potential).
in the comment above we saw the atmospheric window radiation was produced by both the surface and the atmosphere but only warmed the surface. now we have a different type of radiation that warms both the surface and the atmosphere (not directly because the net flow is outward, yada yada yada). if CO2 was not present then the 15 micron IR would be part of the atmospheric window and lost directly to space rather than be captured by the atmosphere. more direct energy loss would result in lower temps for both the atmosphere and the surface. THIS is the greenhouse effect.