I'm pretty sure most people are aware of the huge amount of content available on sites like YouTube, Megavideo etc., and of the existence of hangers-on sites such as Keepvid, Clip Nabber etc. as well that let you leech the FLV files directly. Furthermore, I'm also sure that you've noticed many of these leech sites not working more often than not nowadays. This is mainly due to the fact that the video sites frequently change the code that generate the URL for the FLV file to prevent exactly this form of abuse as it reduces the number of direct hits on their sites thereby reducing revenues and what not.
Depending on your browser, you can still just locate the downloaded file from your browser's cache directory. However, this is something of a pain and generally sub-optimal. The following is an imperfect workaround in that it requires you to load (for a second or two) the video in your browser, but, on the other hand, will always work and what's more, it is not only restricted to video files:
This can also be accomplished by using network tools like packet sniffers and the like.