Video codecs are good at compressing stills, so there's very little overhead in playing such a video when only the audio is desired.

That being said, YouTube does also encode audio-only versions of all their videos; they're not available from the web player though. You can use external software such as youtube-dl[1] to take advantage of it.

[1]: https://github.com/rg3/youtube-dl/ (using an invocation such as "youtube-dl -g -f bestaudio [url]" to retrieve the URL of an audio-only encode)