CO₂ re-radiates energy from high-energy massive (particular) radiation. It is also a greenhouse gas. The two are not mutually exclusive. (The article is misleading in describing CO₂ as a thermostat; it's technically correct, but doesn't apply to this situation.)

When CO₂ is hot, like all things, it radiates more black body radiation. However, it only gets hot when it absorbs things. (Reasonably accurate chart of absorption spectrum.) If you look at the absorption spectrum, you'll see that it mostly absorbs low-energy / high wavelength light – in the infra-red part of the spectrum. This is the band of wavelengths that make up most of what is given off by things that aren't hot enough to glow.

It also absorbs energy from large, non-light particles that rain down on it. (If I shine a bright torch at a clean pane of glass then it won't get too hot, but if I throw glowing pieces of metal at it then it will get very hot – even if I shine more energy at the pane of glass than is in the metal that I throw at it.)

What happened in the solar flare was that lots of high-energy massive radiation rained down on the upper atmosphere, which warmed it up as those particles collided with the air. This caused the CO₂ in the upper atmosphere to glow more brightly (though still not enough to glow in the visible spectrum). Since there wasn't much CO₂ above it to re-absorb the light, the net effect was to re-radiate almost all of the energy dumped into the atmosphere right back into space.

Now, you'll see that CO₂ doesn't much affect high-energy, short wavelength light, like light in the visible spectrum. That's why CO₂ is (almost) invisible; it doesn't absorb visible light or higher-energy light. However, it does absorb lower-energy light, heating it up and causing it to re-radiate that as low-energy light (since it isn't hot enough to glow).

High-energy light (visible, ultra-violet (to an extent; other stuff filters lots of that out), etc.) gets through the atmosphere and lands on the ground. The ground is not transparent to this light; otherwise we'd be able to see people on the other side of the world. (Or whatever's underneath the world; this model works equally well with a Flat Earth. *rolls eyes*) Near-white stuff like snow reflects most light (which is why it takes so long for snow to melt even when the sun is shining directly on it, even though you can melt it easily in your hand), whereas near-black stuff like tarmac absorbs most light (which is why the road is so hot on sunny days). You can test this using a Leslie cube, a torch and a thermometer, or by making your own experiment. Absorbing light makes things heat up, which makes them radiate – but unless they're very hot, they'll radiate low-energy radiation.

Low-energy radiation at the top of the atmosphere will cause CO₂ (and other greenhouse gases, but we're just dealing with CO₂ at the moment) towards the top of the atmosphere to heat up and (net) to radiate heat upwards – towards where there is less CO₂. Low-energy radiation at the bottom of the atmosphere will have the same effect; to radiate heat downwards, where there is less CO₂. The greenhouse gases in the atmosphere act sort of like a conditional mirror; they let through high-energy radiation and "reflect" (but not really reflect) low-energy radiation.

This is actually how a greenhouse works, too; if you get a low-infrared camera and put a cold piece of glass (from the freezer) in between it and something warm (like your hand) you won't be able to see your hand very well. (In principle this should also work with a warm piece of glass, but that piece of glass will be warm enough to be glowing too, so you wouldn't be stupid to blame the absence of a visible hand on the noise / interference.) However, shine higher-energy light (e.g. from a torch) through and you'll see it goes through fine. (Absorption spectrum of Kopp Glass' 3131 filter (HTTP))

I plan to put together a simplified simulation to check my model. It'll be slow, though, since it'll be modelling individual particles of air. It'll also have to have a higher concentration of "ideal greenhouse gas" since my computer wouldn't be able to manage to simulate (6.02214076 * (10 ^ 23)) / 24 = 2 509 225 316 670 000 000 000 molecules per litre (at sea-level / room temperature) of an entire column of atmosphere (stretching tens of thousands of metres) – even if I only simulated the 0.41% of that that's actually CO₂. (So don't say "but there's nowhere near as much CO₂ in the atmosphere as in your simulation, therefore your model is wrong"; unless you can explain why this is a valid refutation you'll have to poke a hole in the actual theory, and not my demonstration-aid inaccurate computer simulation.)