Demo Viewing

With teeworlds made demos can just be viewed with teeworlds.
Otherwise you have to use screen capturing programmes, like FRAPS

Yes, you got it.
Demos are for watching.

Third party programs won't be able to just "convert" .demo files. .demo files are not videos, instead, they are files, which contain all the information of what happened. In the demo file there is something like "player A moves from location (x,y) to location location(x',y') while pointing the pistol upwards..." etc. I don't think there is a program other than teeworlds, that is able to interpret this data. The main reason for this is that it saves a lot of disk space. High quality videos are always big, and the demo files are quite small in comparison. Another pro for demos is, that third party programs are able to extract much more information from the file than from a video (and much easier). Think about tools, that are able to automatically detect the outcome of recorded games, if they are given the demo file (this would be quite difficult with video, because the image of the endgame screen would have need to be analyzed, which is kind of complicated for software).

If you need to somehow "export" the demo to some video format, your only option is using screencapturing software (e.g. FRAPS). You could also use the n-client, which has a built-in demo-to-video feature (as far as I know, this also is just screencapturing).

To conclude: demo files are the best way to record games, and that's the way it's done in other games, too, e.g. SC2, Dota 2, Counter-Strike. However, in other games they are often called "replays", which may be a more appropriate name.