After finding the URL behind a video stream, it has to be added to VLC (as specified in the other answer): Open Network Stream menu item (or Ctrl+N).
To find that URL, here is the link to an older question. The selected answer there is by no means the best. The most voted one is good because it involves just using the Chrome browser. But it does not work with all streams, and neither does the Firebug addon for Firefox, as far as I can tell.
I prefer the Video DownloadHelper addon for Firefox - that I mentioned in my answer there, just install the addon, play the stream, the DownloadHelper button should start rolling, click the drop-down menu, then 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 sometimes like
/video.flv?start=0. Changing that to
/video.flv? might make it work. (But in many cases the stream is played very well without this change. Also, in many cases the link may look completely different anyway.)