OK, I'll paraphrase. Here we have what Earth's average emission spectrum looks like from space, overlaid with ideal black body emission curves at different temperatures, alongside the relevant portion of the measured absorption/emission spectrum of CO2:
Now, the area under the curve on the left (well, if you continue it off the left at least) defines the total amount of energy Earth releases to space per unit time. So, anything that reduces the area under the curve will increase the temperature at ground level.
Look at the great big dip in the middle (helpfully labelled "CO2"). If you match up the numbers on the bottom axis with the blue graph on the right, you'll see that the shape of this dip matches quite closely to the (flipped) CO2 spectrum.
The key insight to grasp here is that the effective emission temperature seen from space for any given wavelength is the temperature of the height in the atmosphere where whatever has been absorbing/emitting the radiation on the way up loses the ability to do so any more (usually because there's simply not enough of that species above that height). For water that's particularly obvious when you look at the left-most part of the graph: see how the emission intensity hovers between the 260K and 280K blackbody curves? That's because the water freezes out at 273K - above that mark in the troposphere the atmosphere is essentially transparent in the water bands. CO2, on the other hand, doesn't freeze until a little below 200K - so it's effectively always there, and only stops blocking in its band once the atmosphere itself becomes so thin that there's not enough left to get in the way. So it pushes the effective emission temperature way down into the vicinity of 210K (-63 Celsius) at what would otherwise be Earth's peak emission band. Now, the centre of that dip is more-or-less saturated, but adding more CO2 will *widen* the "shoulders" of the dip, inevitably reducing the amount of outgoing radiation. The only way for the Earth to come back towards equilibrium is to warm up until emissions in the other bands increase to compensate.
