As seen from the many answers there must be many ways to do this, but maybe one of the easiest (especially for Firefox users) is the DownloadHelper addon, which has also the advantage of being available on all OSes.
- Install the Firefox addon DownloadHelper,
- run the flash stream in Firefox, the DownloadHelper button should start rolling,
- click the drop-down menu,
- put the mouse over the title of the video and then choose 'Copy URL'.
Testing such links with VLC I notice that in some cases the player cannot play them or even crashes. But this can be fixed as follows:
The last part of such URLs is something like
/video.flv?start=0. Changing that to
/video.flv? might make it work.
I discovered this when using the other solution, presented below, Internet Download Manager, in order to capture a such video stream: it displays the link but when trying to select it by clicking on it, IDM selects the URL without the last part
start=0 (as seen in the image below). But many URLs can be opened in VLC with that last part too.
In Windows, Internet Download Manager is able to sniff such streams and capture them. Again, I find it easier to use in combination with a Firefox addon, FlashGot, but IDM can also be used alone with any browser. When ready to get the video, IDM displays the URL:
Considering the edit:
"Edit: The player is streaming an RTMP stream, not an FLV video file."
RTMP (Real Time Messaging Protocol) uses either FLV or MP4 as video format.