This is a wonderful TED talk by a climate modeler named Gavin Schmidt. He explains what goes into the million line code that is the modern climate model in terms of the science, and, most importantly, how climate scientists go about checking to see if the models are working. The method: take real data and trends, run the model with the appropriate initial conditions, and simply see if the output of the model looks anything like the real-world. This is how any type of simulation experiment works. When the model begins to become statistically consistent with real data, the model is called 'skillful.' Climate models have become skillful, as shown in this presentation.
What is really cool about having any type of reliable, tested, skillful model and simulation is that one can then make predictions. In the case of climate models, the user can put in any conditions and tweak as many parameters as they want, run the model over time, and see what the world will look like under those conditions. He shows this, as well, for three cases: cutting back on carbon emissions a little, cutting back on carbon emissions a lot, or doing nothing at all with things happening at current rates. What the world looks like comparing those three scenarios is eye-opening.
And, keep in mind that as scientists have fine-tuned the models using past data (and they go back tens of thousands of years), and tweak parameters based on the measurements, the way they get a world that looks like today's is by adding in the carbon emissions over the past century.