Note the 'Download' link, too....
Note: after downloading, you'll get asked to 'Register' with the DRM rights holder. And the process may fail. The trick seems to be to make sure your phone is already online before saying 'Yes'. In my case, I started up Web on a web page first.
Downloaded BBC programmes then play fine! Incidentally, the timing of this release leads me to suspect that it was yesterday's v20 firmware update that enabled the appropriate DRM handlers in the 5800 to make the full iPlayer functionality possible. Anyone still on v11 firmware should note that the 'Download' bit won't work.
(Thanks to Michael Reynolds for the heads-up)