A greenhouse stays warm because its glass walls and roof allow heat to come in via radiation from the sun but prevent the loss of that heat due to atmospheric convection. The atmosphere itself is transparent to the bulk of the radiation we get from the white hot sun, but some of its components (most significantly the water vapour) absorb a significant proportion of the infra-red radiation that is emitted from the Earth’s cooler surface. What is absorbed gets re-radiated in all directions, with essentially half eventually continuing outward and half returning to the Earth; and this has the effect of raising the surface temperature for the equilibrium situation at which equal amounts of energy flow up and down at every level (including both the ground and the top of the atmosphere). Because the atmosphere allows heat to reach the ground but acts as a (partial) barrier to heat loss, this is generally referred to as the atmospheric greenhouse effect (even though the mechanism of heat flow blocking is of course different from that of an actual greenhouse).
In the absence of any atmosphere, the temperature at the surface of the Earth would have to be such that
The same applies at the top of the atmosphere (ie at any level above which there is no significant amount of further atmosphere to block outbound radiation)
If we imagine an atmosphere which allows all incoming radiation to heat the Earth and absorbs a percentage of the outgoing which it then re-radiates, then if we also imagine it as having a uniform temperature we find that the temperatures are related by the following equation:
For the case of an atmosphere of uniform composition which absorbs a fixed percentage of the outgoing radiation per unit of atmospheric mass and with density decreasing with altitude according to the usual ideal gas law we can, with a bit more work, obtain the temperature profile as follows:
But in the Earth’s atmosphere the composition is not uniform because the percentage of water vapour depends on the temperature
Also, the absorption is restricted to specific frequency bands so the total absorption percentage can never exceed the percentage of the spectrum that corresponds to those bands
an atmospheric layer that is almost totally opaque to the absorbed frequencies will not in fact completely eliminate them from the Earth’s emitted radiation. This is because when heated up by absorbing that radiation, the layer in question radiates a full thermal spectrum from both its top and bottom surfaces and that from the top includes a portion that could be absorbed by a second such layer – and so on. So the number of such layers matters and increasing CO2 does lead to a slightly increased surface temperature (but the effect of successive layers diminishes so that the overall expected warming is only (approximately) logarithmic, with successive doublings of the CO2 concentration each giving roughly the same small temperature increase). But even just a “small” increase can have a big environmental effect – especially if it is unusually rapid in geological terms and relative to the capacity of species (and economies) to adapt.