OK, I got this to work.
It looks like this system uses a form of authentication for each stream user, though I'm not sure what the point is.
On the rapidfireradio.caster.fm site, press the second item on the right side of the gray bar "Rapid Fire Radio" (the tooltip says "Tune in with your media player", it's a small triangular play button in a rectangle).
Then scroll the page to expose the "Tune-In Links Are Unique to You, You Cannot Share These Links." section, and the URL is at the bottom of the white box, labeled "Direct Link To Be Added Manualy" (sic) - it looks like:
...where the "blahblahblah" is the authentication code, which is apparently required and unique to you. This need to be entered *exactly* into the URL section of the stream on the device, which is tricky given the complexity of the URL and the limitations of the on-screen keyboard.
To avoid entering this whole thing on the device and risk error, what I did is was create an entry with no URL on the device, then SSH'd into the device and used "vi" to edit the "/psp/url_streams" file to insert the URL, synced the file system, then restarted the Control Panel (or you can reboot).