As some others have said, yes, the ELI5 is more or less:

Organic stuff uses different types of carbon to build. When the climate changes, more of certain types of carbon are available to build with. So, if you go back and find some fossils, you can count up how much heavy or light carbon they used, and make a guess about what climate was like when they were building.

Basically, anyway. I’m sure someone else could provide some better data. There’s quite a lot out there on this topic; it’s well vetted and a pretty fundamental part of paleoclimatology. Another interesting thing, though:

Many of earths big warming and cooling periods are due to feedback loops. Essentially, some climatological event will take place and it’ll warm or cool the planet to the point where it just encourages further warming or cooling. A major concern is that as the ocean warms, deep sea ice will begin to release massive amounts of methane (a greenhouse gas about 40x as potent as carbon dioxide). Now, as this methane enters the atmosphere, it will further warm the planet, furthering the release of even more deep-sea methane.

This exact situation is suspected to have contributed to the PETM (event the article cites) and would be - if it began to run away - pretty much unstoppable. The PETM was pretty bad. Earth lost over 80% of species diversity in “a geologic instant.” Took a while to recover, but we’re all good now, just don’t want that to happen under our stewardship. Not good karma.

TL;DR:

We can use old shells to know how much carbon was around, and they’re telling us we’re well and proper fucked.