The streaming service is working just fine. When enabled, it's accessible from any device connected to the IP subnet your moOde player is on, just as the moOde Web-UI is. By default, the UI is served via port 80 while the streaming audio is served via port 8000. These are "well known" port numbers for these services.
<net-nanny hat on>
Please, oh, please, even if you read the manual for your router and figure out how to forward either of these ports to the Internet (and set up dynamic dns somewhere so you can find them by name), *don't do it!* Especially port 80. As Tim said, moOde is not a secure platform. (This is not a problem unique to moOde!)
The Internet is a wild and woolly place these days. The black hats out there have automated software scanning virtually the entire Internet 24/7 looking for targets of opportunity (you can see it in firewall logs) and an open port [1] is a shining beacon to them. To my way of thinking, the risk of having your RPi become their penetration-testing device on your LAN outweighs the dubious benefit of listening to a stream from your moOde player while away from it.
While I'm at it, please don't forward any UPnP ports from any device on your LAN either. UPnP is notoriously insecure technology.
<net-nanny hat off>
Regards,
Kent
[1] You can visit Steve Gibson's Internet security vulnerability profiling services at
https://www.grc.com/x/ne.dll?bh0bkyd2 if you are curious about how your network looks from the outside.