The proof that man-made CO2 is causing global warming is like the chain of evidence in a court case. CO2 keeps the Earth warmer than it would be without it. Humans are adding CO2 to the atmosphere, mainly by burning fossil fuels. And there is empirical evidence that the rising temperatures are being caused by the increased CO2.
Addendum : the opening paragraph was added on 24th October 2013 in response to a criticism by Graeme, a participant on the Coursera Climate Literacy course. He pointed out that the rebuttal did not make explicit that it was man-made CO2 causing the warming, which the new paragraph makes clear. The statement ".and humans are adding more CO2 all the time" was also added to the 'what the science says section.