I want to be able to listen to audio e.g. songs by streaming (via airplay) from YouTube video to my apple tv that is connected to my audio receiver via the optical cable. I don't want to keep my TV on in order to listen to songs from YouTube video via apple tv.
Apple TV
You could do it on a Mac by using the AppleTV as the audio output setting rather than using Airplay Mirroring if that's how you were doing Airplay.
Hold the left alt/option key while clicking on the speaker icon in the top menu - available audio out destinations are seen here:
Audio Airplay capable devices are listed with the Airplay icon as in the example above.
This is different to Airplay mirroring which may need to be disabled as it sends both video and audio together.
Thats not possible, it's one stream and you cannot separate audio from video. If you want to listen to just audio use AirPlay with a music app
how can I airplay only the audio to apple tv from a youtube video
