Often this is accompanied by some explanation, something along the lines of "I mean, a teleconverter surely degrades the image, so why bother? Why not just shoot without the teleconverter and crop?"
In brief, the reason is that while the image shot with the teleconverter will look less sharp, it's quite likely to show more detail on subject.
That may seem like a conflict, but it's not.
Contrary to popular myth, at this time many good lenses actually out-resolve most digital sensors. What the teleconverter does is simply to make the optical image bigger. This process enlarges the image's original flaws and adds some more of its own, but the enlargement also allows the sensor to capture detail that was previously inaccessible because of the sensor's finite pixel size.
Here's an example. These were shot with a Canon T1i camera (4752 pixels in 22.3 mm = 4.69 µm pixels), using a Canon EF 100mm f/2.8L Macro IS USM main lens, with and without a Kenko C-AF 1.4X Teleplus Pro 300.
For context, here are the full frames. We'll be looking closer at that resolution chart taped to the door.
Main lens by itself:

With teleconverter added:

A closer view, main lens by itself at 100% pixels, stepping aperture from f/2.8 to f/32:

With the 1.4X teleconverter, at 100% pixels, f/4 to f/45:

And even closer, main lens by itself at 200% pixels:

With teleconverter, at 200% pixels:

You can see in these images that the setup with teleconverter resolves about two more sets of bars. There are six sets per 2X, so this means that the teleconverter has increased the effective resolution on subject by about 26%.
If both the teleconverter and the lens were perfect, we might expect an improvement in proportion to the power of the teleconverter, about 40% in this case. Presumably the missing 14% is due to imperfections in one lens or the other, and the experiment doesn't tell us which.
You can also see in these images the subtlety of cambridgeincolour's calculation that for a 15 megapixel APS-C sensor, "diffraction limits standard grayscale resolution" at f/10.7. Sure enough, at f/11 in these images, there's a slight but unambiguous loss of resolution on the finest bars. Nonetheless, the images with teleconverter show that the f/11 optical image resolves at least two more blocks of bars that were missed by the camera. There's plenty of blame to go around in this case -- at f/11 the lens is not fully exploiting the sensor, but also the sensor is not fully exploiting the lens!
Anyway, the image with teleconverter is 40% bigger and shows 26% better resolution on subject. The missing 14% makes the TC image look less sharp, even though you can see finer detail on subject.
I hope this helps.
--Rik