How scientists know what earth's climate was like in earlier geologic eras?

Answer 1

They use climate proxies to infer characteristics of climate in the past. For examples, atmospheric gases become trapped in antarctic ice. Scientists can now drill into the ice and take a core sample, testing the ice to determine what the atmosphere was made of, and therefore how it behaved, in the distant past.

A:

One component, the temperature, of the archaic atmosphere may be determined by the ratio of two isotopes of oxygen. O16 and O18. Their contribution to the atmosphere is a function of temperature.

And in the very early Earth, there was little oxygen therefore there was little photosynthesis going on.

And in the late Carboniferous Era, the oxygen content rose to over 30%, indicating massive photosynthesis and a warm climate. In this era, some dragonflies grew to have a 75cm wingspan! Only possible with plenty of food and a warm climate.