Let’s keep this short.
Facebook made a change to videos posted from YouTube, Vimeo and iPlayerHD among others.
Until the change, a video posted on a Facebook page from third party sites played on the Facebook page. Now, videos posted from third party sites open to a page hosted by the third party site.
It’s not a stretch to understand why Facebook made this change. Facebook has no way to monetize videos hosted by third parties like iPlayerHD and YouTube. They hope you will upload your video to Facebook to keep the video playing on Facebook so they can monetize (make money, pay the bills) your video.
The good news is Twitter has not (yet) made this change and we have no knowledge of their future plans on this issue.
Have a nice day.