The extension can be accessed by tapping on the "+" icon next to the text bar to add Apple Music to a conversation. There's also an option to interact directly with the Apple Music chat bot to ask for music recommendations.
Apple Music subscribers can listen to full tracks from Apple Music, while non-subscribers are able to share and listen to 30 second clips from any music track. There's an option directly within Messenger to sign up for Apple Music.
This feature appears to still be rolling out to users. While Engadget has access, we're not yet seeing full functionality.
Facebook first announced plans for both Spotify and Apple Music integration in Messenger in April of 2017. The Spotify chat extension has been available for months ahead of the launch of the Apple Music extension.