Once you've linked Apple Music with Alexa, you'll be able to set Apple's streaming service as your default music source so Alexa will play songs from it by default. Alexa will have access to your personal library and playlists that you create and offers playback controls while you're listening. The voice assistant won't be able to add songs to your library or play music that you've added manually from your own music collection.
The support on Echo devices marks only the second voice-controlled speaker to work directly with Apple Music, other than Apple's own HomePod. It's not clear if other Alexa-powered speakers like the Sonos One will offer similar compatibility, though a blog post from Amazon makes it sound like it's limited to Echo speakers only.