Youtube videos are embedded via FileViewers, which are not rendered by default within content served from REST. This is an area we are looking at improving. However, as a workaround, you can set un undocumented parameter to your REST request, 'PostTarget'. By default, the target is 'WebServices'. However, if you pass the value of 'Web' for 'PostTarget', it will format the body the same as it would be when rendered within the widget, including FileViewer content transformations.
Hope this helps