There are two directions to look at it: convincing yourself via a convincing story, or convincing yourself via a convincing calculation.
The story they tell defies basic thermodynamic intuition, and seems to skip over all the parts that would put the narrative in doubt.
The premise is that greenhouse gases absorb black body radiation from the surface.
First of all, greenhouse gas is a bad name, greenhouses warm because of air convection. So the "greenhouse" intuition they try to abuse is misleading.
Secondly, think about this thought experiment: a layer of pure CO2 gas covers the lowest layer of air near the surface. By how much would that get warmer compared to air? The answer is there won't be any difference. That's because the surface and the air near it are at the same temperature, so they emit exactly the same amount of energy. All systems with same temperature are at an equilibrium.
So what can possibly cause warming is only the effect of the difference in temperatures between the CO2 gas and the surface. Which should only be significant in higher altitudes. Which have much less CO2 because it is heavier. And it's also not clear, in this situation, in which direction the weather near the surface will change.
But then you realize there are also effects of increasing methane / CO2 which decrease warming. The most obvious effect is that they have higher heat capacity. Heat capacity goes together with absorption: the absorption is possible only because the molecules have more degrees of freedom to vibrate. So it's usually around the same order of magnitude. And higher heat capacity means faster cooling from convection and it means the air requires more energy to heat up. In fact, you can think of the entire climate change claim as a statement about the entire earth heat capacity.
So I'm entirely unconvinced by the story. But that should've been OK because there are detailed calculations. There are models.
Except these models are dumpster fire. You can download some of them from NASA's website and judge the quality of code for yourself. Old fortran code, all the models copy code from each other, many things which aren't constant physically are constant in the code, functions full of tens of "if else" statements whose physical validity is highly in question.
But those models should've been tested? Except they don't. There's no way to test them. From basic software engineering perspective, it is insane to trust these things to make the kind of trillion dollar decisions they make.
But all models point to climate change! Well they all copy code from each other. But that's not the only problem. They are all thermodynamic simulations which substitute the full state of the system with average states.
In other words, they all, by design, underestimate the entropy of the system. And thermodynamic energy is entropy times temperature, and since they get the energy right (just the sun) you expect them all to get higher temperature. So it is actually expected from all simulations to overestimate warming.
There are also other glaring counter intuitive things about the narrative. Somehow CO2 effects continue to work slowly over decades, and the system doesn't reach an equilibrium with the current levels of CO2 until decades ahead.
This is completely against normal thermodynamic intuition. Meta stable states exist, yes, and there are out of equilibrium systems. But usually the reason for systems not reaching the more stable state, is that their fluctuations are too small. In this case, the fluctuations are much much bigger than the supposed stable state change. We're talking about barely a degree change over decades in systems that fluctuate by several degrees daily. It makes no sense to claim their equilibrium will only be reached in decades while their fluctuation exceeds the difference to equilibrium daily.
