Custardo, what we know is that it's harder for the Earth to radiate off heat the more CO2 there is in the atmosphere.
That's because the Sun is very hot, so gives off relatively high frequency light, the sort you can see with.
The Earths surface almost never gets hot enough to glow in visible light, but does so with lower energy light we label infra red.
We evolved to see in the wavelengths that the air is transparent to. Bit it is less transparent to infra red, and co2 is a sort of fog, so heat leaves the Earth more slowly the more you have of that sort of gas.
That's a straight bit of physics you can do in a school lab.
Humans have been burning carbon based fuels a lot in the last few centuries, and at the same time CO2 has gone up. A lot.
But that's not proof that humans did it. Lots of other sources of CO2, indeed human activity is a relatively small % of CO2 produced on earth. If humans stopped burning carbon, it would be sucked up by plants quickly.
A change in the balance between CO2 producers and consumers could produce a far bigger effect. We know this has happened in the past, though it's hard to get an accurate timescale on how quickly the change happened. It's believed that it took longer, usually.
We know that for a large % of it's history the Earth has been a lot warmer than it is now. Many scientists call the time we are in now an ice age, because there is all year round ice at the poles. That has often not been the case. The use of "ice age" has of course been a casualty of political correctness, and is in rapid decline.
The Earth is strongly homoeostatic, ie it will act to reduce stimuli applied to it in accordance with Le Chatelier's principle. When this was rediscovered and applied to complex biological systems by Lovelock, dippy green arts graduates by the million thought he had discovered the Earth Goddess.
They get a bit sad later when since he was proper scientist he supported nuclear power.
For instance the atmosphere is a gas, when you make it hotter it expands. This uses up some heat, and also increases some of the ways it can lose heat. Heating/cooling water is more complex, but vital since such a large % of the Earth is covered by it. Water vapour is itself a greenhouse gas, but clouds reflect incoming light.
Almost all plants don't get enough CO2, so if you increase the supply they grow more, sucking it out of the atmosphere faster.
Thus we have been able to get away with seriously dumb stuff like burning oil, growing rice and raising sheep, without trashing the atmosphere. But since only the first is seen as an American activity, it gets the most focus by the arts grad Greens.
CO2 does seem to correlate awfully well with both increases in temperature, and more dangerously a more variable set of temperatures.
Thus CO2 is like smoking. The first evidence of it being bad was entirely correlations, we simply did not know enough to work out what was going on. Even now meedja types still give publicity to those who claim that passive smoking is safe, and for many years used the term "nazi" to anyone who implied smoking was a bad thing.
The test for any scientific model is predictions. If you feed the model data it has not "seen" before, and it gives numbers that are like what you see in real life, then it's a good model. The more accurate the numbers ,the better the model. Our climate models are quite good on this score, but what you won't see on the BBC is any critique or health warning about computer models. Part of my work includes this, and right back when I started, the example of computer models giving mad results was climate ones. They produce such strange numbers sometimes that the whole field of Chaos theory evolved from trying to work out what's going on.
We've come a long way from that, but computer models do not equal truth, a fact that a Apple Macintosh using arts grad from the BBC can no more grasp than the fact that 11 samples are not a valid basis for saying that MMR is dangerous.