How do I cast from YT Music to ChromeCast Audio?

As of now, you can't. Google does not support integration of YT Music and Chromecast Audio.

For now, you'll have to use the Chromecast App sound-transcoding. You can find those instructions here, from your Android device:

1. Connect your Android device to the same Wi-Fi network as your Chromecast Audio.
2. Open the Google Cast app Chromecast cast icon.
3. In the top left corner, tap the navigation drawer More menu icon.
4. Tap the Cast Screen + Audio button.
5. Select your Chromecast Audio.

This has a lot of caveats. The sound is inferior. The device that is doing the screencasting is communicating to Chromecast Audio directly (putting your hand over the antenna and killing the wifi throughput will cause skipping). And, you won't be able to answer a phone call.

With the release on September 1, 2016, after wayyyy too long this was fixed. You can now cast as you normally would to a Chromecast A/V device.

Another solution that is easy to start:

1. Go to youtube.com/your_clip and click the normal Cast Button
2. In the dialog box Select source

3. Select Cast tab, and then the Audio Cast option will be available: