The greenhouse effect analogy of global warming is that atmospheric carbon dioxide CO$_2$ absorbs some of the infrared radiation emitted by the Earth, and redirects a portion of that radiation back down to the Earth's surface, thereby heating the surface more than it would have done if that radiation had been able to escape into space.
Global warming is then simplistically explained to the general public by the idea that as atmospheric CO$_2$ concentrations rise, more infrared radiation is absorbed by CO$_2$ and re-emitted back down to Earth, causing increased heating of the Earth.
However, this explanation is not technically correct, because at present atmospheric CO$_2$ concentrations, just one kilometer of atmosphere is sufficient to fully absorb all the infrared radiation emitted by the Earth, at the wavelengths at which CO$_2$ absorbs.
Carbon dioxide absorbs infrared at the wavelengths of 2.7, 4.3 and 15 µm, and the CO$_2$ in the first kilometer of atmosphere alone is able to completely absorb all infrared at these wavelengths.
So the infrared absorption process is already fully saturated, and thus further increases in atmospheric CO$_2$ will not lead to any additional absorption. This is why the simplistic explanation provided for the general public does not seem to be technically correct, even though it roughly outlines the idea.
I found one blog article by Clive Best that tries to explain the actual process behind CO$_2$'s ability to cause global warming. Judging from that article, the actual process is more complex than the simple explanation provided for public consumption. However, I don't fully understand the explanation given in the article (and from what I did manage to understand, I am not sure if it is fully correct).
So I wonder if anyone here can provide an easy to understand explanation of the actual mechanism by which increased atmospheric CO$_2$ leads to global warming. Or perhaps if you know any good articles that explain it, please can you post the links.
I tried to find some info on the actual mechanism of global warming via Google, by using search terms such as "mechanism of greenhouse effect in global warming", but was surprised to find very little information available.
I also asked this question on physics.stackexchange.com here.