How do we know the rate of weather change from millions of years ago just from fossils?

Info doesn’t just come from fossils. Geology is used as well, ie; differing layers of sediment, rock, even layers of fossilized burned vegetation, flooding or lava flows form a basis to be able to judge earth’s history.

I assume that the further back we go in time the more inaccurate we are on how the climate change and we can only know about the general climate from the plants and animals we find.

That is why the older something is, the wider the scientific dating can be, ie: managed agriculture methods began around 12,000 yrs ago vs horses developed 45 to 55 million years ago.

Also why didn’t you question OP’s source of data?

Because the info OP gave is accepted science and can readily be found everywhere.