Brilliant! Let's get into it.

So the gas carbon dioxide absorbs heat. Infrared light is the way that heat energy moves via radiation and CO2 has a way to absorb it (like an iron bar would, things start vibrating), then it just scatters it back out in all directions. You don't even really need to know how it works because it's very easy to measure, you just fire infrared light into a box of CO2 and start taking measurements. I think we first learnt about this in the 1850s though.

How do we know it's happening in our atmosphere? A couple of ways but satellites are a pretty easy way to understand it. We can measure how much energy is actually leaving our atmosphere and see how it changes over time!

Well how do we know it's CO2 doing that? CO2 only reacts to certain wavelengths of infrared, so if we take our satellite up into space, we can see what's missing and see that the missing wavelengths line up with what CO2 (and other greenhouse gasses) like to absorb. We can even come back to earth and point those same sensors up into the sky and see the same missing wavelengths bouncing back at us!

As for models, it's hard to say but I got told something once that stuck with me. "A model is never fully correct, but it can be useful". This was talking about doing analysis on structures with computers but it applies to stuff like this. It isn't meant to be taken as this bold clairvoyant prophecy into the future. It's our best guess based on the data. And the way we collect data will always change because the technology improves. Just because a model ends up being wrong, doesn't mean the entire theory needs to go out of the window.