Physics (specifically radiation transfer mechanics) provides the compellingly supported understanding that CO2 is capable of acting like a "greenhouse" gas, the PETM is nothing more than a recent geological example of what happens when you rapidly (over the period of ~ 10k years) flood the Earth's atmosphere with climatically significant volumes of such greenhouse gases. We are currently adding CO2 to the atmosphere at ~10x the rate that occurred during the PETM and our rate of emissions is still increasing.
And what happened during the PETM? I'll let wiki explain it....
A bit of topic stray, but definitely relevent and even essential to the proper understanding of why AGW topics are more than just a political debate in disguise. Wiki is problematic in itself, not that it is not handy for messageboard reference, but as a primary source it is never any better than the 10s to 100s of often "non" and "un"qualified contributors that happen to shape the information on its pages. It's a good place to start searches when you are looking at a topic, but it is a very poor place to end a search when looking at any topic. That said, we can look at what it says...
Technically, these radiations occurred as climate change killed off, or back, the dominant species successfuly competing and managing within the existent ecosystems. As these dominant species are unable to demonstrate the same dominant success in the changing environments. Species that can more rapidly adapt gradually replace those that can't and variate to fill vacated niches. Any time you see points of rapid evolutionary radiation, you are looking at periods of time where some force or factor is stressing the environment. Mild, temperate, stable, some might even use the term "comfortable," environments are rare in the geologic history of our planet.
The deep-sea extinctions are difficult to explain, as many were regional in extent. General hypotheses such as a temperature-related reduction in oxygen availability, or increased corrosion due to carbonate undersaturated deep waters, are insufficient as explanations. The only factor global in extent was an increase in temperature. Regional extinctions in the North Atlantic can be attributed to increased deep-sea anoxia, which could be due to the slowdown of overturning ocean currents,[12] or the release and rapid oxidation of large amounts of methane.[20][verification needed]
In shallower waters, it's undeniable that increased CO2 levels result in a decreased oceanic pH, which has a profound negative effect on corals.[21] Experiments suggest it is also very harmful to calcifying plankton.[22] However, the strong acids used to simulate the natural increase in acidity which would result from elevated CO2 concentrations may have given misleading results, and the most recent evidence is that coccolithophores (E. huxleyi at least) become more, not less, calcified and abundant in acidic waters.[23] Interestingly, no change in the distribution of calcareous nanoplankton such as the coccolithophores can be attributed to acidification during the PETM.[23] Acidification did lead to an abundance of heavily calcified algae[24] and weakly calcified forams.[25]
The increase in mammalian abundance is intriguing. There is no evidence of any increased extinction rate among the terrestrial biota. Increased CO2 levels may have promoted dwarfing[26] – which may have encouraged speciation. Many major mammalian orders – including the Artiodactyla, horses, and primates – appeared and spread across the globe 13,000 to 22,000 years after the initiation of the PETM.[26]"
This section is extremely shaky and filled with much weakly supported speculation rather than the more general and compelling mainstream understandings. Broad and general "die-backs" need only a few significant extinctions at the base of foodchains to make major changes in existant biosystems. Unfortunately, many people tend to perceive "extinction events" as only occurring when there are huge net losses of species; this understanding excludes most natural extinction events where the actual net loss of species is low as you have high losses of species being offset by high levels of radiation in other species.
This is more common in natural climate change events which are gradual enough that mobility and natural variation can filter specie traits enough to encourage adaptation and evolution. Fortunately, neither adaptation nor evolution will allow a lobster to accomodate the cook-pot.
So, in a nutshell, certain species of forams suffered very high extinction rates. Different species on the other hand did very well. Mammals did exceptionally well and contrary to the incessant nonsense about heat killing the opposite is true. Warmth allowed plants to grow well and that allowed fauna to do well.
How do you explain that?
Paleocene
The issue isn't so much "how", but rather, "what" it is you expect me to explain? I don't know where you get the idea that I or anyone else here (with possibly one or two apparent exceptions) is arguing that AGW is going to turn the planet into Arrakis, though spice sounds rather nice! What we are saying is that in an era where humanity has already had dramatic ecosystem impact and instituted planet-wide die-backs and die-offs, the additional environmental stressors brought on by AGW climate change should not be ignored nor discounted. Likewise adaptation and motility advantages are more likely benefit species we consider pests and nuisances than species we prefer for pleasure and profit. If viable open-air agricultural lands decrease significantly our own species may well be one of the ones in die-back, but frankly, I'm more concerned about the long term economic impacts to our society and culture, than I am the reversal of our specie's population explosion.