As there has been no sign of a Fermi explanation so far, I had a go for myself, largely with the help of this paper. Yes, I am aware that Lindzen is widely held to be a climate change denialist and therefore an evil monster, but I admire the way he writes clearly and convincingly and without belittling or talking down to his audience.
The concentration of carbon dioxide has increased since the beginning of the industrial age from about 280 parts per million (ppm) to about 390 ppm. According to the simple 'textbook' model* of global warming, it is generally accepted that the warming from a doubling of carbon dioxide would only be about 1°C. That is, the amount of carbon dioxide currently in the atmosphere (about 390 ppm) would have to double to 780 ppm in order to cause a rise in global equilibrium temperature of 1°C. Simple arithmetic based on the rate of increase in CO2 (see the graph above) suggest that this would take almost 250 years to happen. Furthermore, because of the logarithmic effect - each doubling producing the same arithmetic increase - it would take 250+500 = 750 years to produce a 2°C rise. This is hardly catastrophic, and gives us some time to do something. I'll hang on to those incandescent bulbs for now.
But this is much less than current climate models suggest from the warming from a doubling of carbon dioxide. Why is that?
*To avoid any confusion, I mean the simple textbook model described here as ‘this model’, i.e. where it says “Thus this model predicts a global warming of ΔTs = 1.2 K for a doubling of carbon dioxide.”.