This is basic physics. ANY thing will radiate, even much cooler than that. The greenhouse effect works because so much of Earth's thermal emission is coming from cooler temperatures in the upper atmosphere. Without absorption in the atmosphere, radiation would all just come straight from the surface, and we'd have the cooler temperatures down at the surface as well.
Skyhunter gives a good first order summary... greenhouse effects mean that much of Earth's thermal emission comes from high in the atmosphere, rather than the surface.
It's a bit more complicated than a single "emission level", however, because the atmosphere has an "infrared window" from which you get some surface thermal emission, unless blocked by cloud, in which case you get the thermal emission through this window from the top of the cloud layer.
But look at the emission spectra given above. This is a pretty basic observation, and you can clearly see the temperatures at the bottom of the large chunk cut out by greenhouse effects. It corresponds to radiation at about 220K -- which is about -64F.
The window around 8 to 13 microns radiates at about surface temperatures, apart from a small chunk with ozone absorption.
The longer wavelengths show lots of H2O absorption, and hence the emission to space comes from lower in the atmosphere, where there is most of the water vapour. Here the emissions correspond to temperatures of about 240K up to 260K as you approach the big greenhouse chunk... and that's around -28F to 8F.
ANY temperature will radiate. Even a lot less than -60F. And the whole POINT here is that the greenhouse absorption of thermal radiation lower in the atmosphere means you get the cooler radiation getting out into space.
The Earth's total radiation averages out to about 240 W/m^2... which would be the same as a blackbody at a uniform 255K, (-18C, or -1F).
If there was no thermal absorption in our atmosphere, all that emission would be coming from the surface, and Earth's surface would be a chilly -1F, on average.
Cheers -- sylas