One option is to try the Live HTTP Headers Firefox addon. Open the plugin, load the page and start the video playing.
Somewhere there will be the request for the video file, usually with the extension
It can help to save the headers to a text file, then open it in a text editor (so you can use it's text-search)
Another option is to let the video fully load, then prod around Firefox's cache directory for the file. "How To Save Flash Video From Firefox Cache Folder" explains this well:
Wait for the flash video loading completely, i.e. able to play MTV to the end
Open a new tab / window of Firefox and enter about:cache as an URL
Take note of the Firefox cache folder path specified in Cache Directory: and browse to this folder with Windows Explorer
Sort the cached files by file size and try to open the biggest file with Storm Codec-MPC or FlV Player. (The biggest file is likely to be the cached YouTube video)
OK. If you want to know what those files in the Firefox cache folder exactly are, here we go:
While in the about:cache page, click the List Cache Entries link of the Disk cache device section
Press CTRL + F to search for Key: (URL) with the get_video keyword and also with the biggest Data Size: (YouTube doesn’t cache the flash video in FLV extension. For other flash video hosting site, you may save time by trying to search for .FLV extension).
Right-click the URL and select the Open Link In New Tab and refer to the File on Disk: