My favorite app, for YEARS, in this situation has always been Swiff Player:
Swiff Player
For Firefox, the file will be located here:
C:\Documents and Settings\
user name\Local Settings\Application Data\Mozilla\Firefox\Profiles\
profile.folder\Cache
For IE, the file will
probably be located here:
C:\Documents and Settings\JPB\Local Settings\Temporary Internet Files\Content.IE5\
folder
you might have to hunt through a few folders for IE files...you also won't be able to browse to content.ie5--I usually just browse to the "temporary internet files" folder and then append
content.ie5 to the address bar up top. In IE, the file names seem to be prepended with
get_video, so you could search for that string and you should find a bunch.
Anyways, just snag the file and play it! If it doesn't already have the extension, then append
.swf to it and you'll be good to go. The file you want to pick out will probably be at least a few megs in filesize. If you're new to hunting for arbitrary files, it might be easier if you delete all temporary internet files first, and then just go straight to youtube and view the video you want...then watch the folder for a file that increases in size (as it's downloading) and that'll likely be the one! You can also sort by date and have the newest at the top, but if you want to grab older files, you'll have to learn how to distinguish the right files by finding a few first.
edit: ok...I actually tried to do it with a youtube video and the files wouldn't play for me, but Windows hasn't been acting up to par lately for me, so maybe it'll work for you. Just wanted to let you know that so that you don't get your hopes up. Maybe the files are encoded with something odd...I'm not sure yet. I'll keep messing around and double-post if I find anything useful.