It appears that the new ‘Prevent cross site tracking’ security feature on Safari (and perhaps other browsers and devices if you are having trouble) is what is causing embedded videos from displaying.
By default, Safari doesn't accept cookies from cross domains making the use of iFrames with embedded content problematical. This is especially the case for videos, as Google uses the YouTube player and authorization cookies to provide access to this content. Unfortunately, this can only be solved by disabling the 'Prevent cross-site tracking' option on the device that is playing the video.
If you want to remain secure for the rest of your browsing, simply disable the feature (in the browser’s preferences) while you visit this site, then reenable it before you go somewhere else you don’t trust…