FBDown offers a way to download videos from Facebook without requiring your username or password.

FBDown will be of use for those times when you see something cool or interesting while scrolling through Facebook (it happens) and would like to keep the video for watching at a later time. FBDown is a small and unobtrusive video downloader that will get the job done in minimal steps. It does have a few options where you can choose to download with no Facebook account, with a Facebook account, and finally cookies. This last option will help if a video was removed, you can add the cookie file to successfully download.

Keep in mind that not requiring credentials will only be available if the video you wish to download is public, others will require your details.