I had this issue and struggled with the technical pieces of it but was able to follow the instructions at this link, https://developers.google.com/youtube/registering_an_application.
I created a new project, enabled YouTube Data API v3. Then I did a OAuth 2.0, and finally the API Key. I deleted the referrer restriction, and I actually regenerated the API key, and copied it into sidebar widget info area. Then it worked!! What a relief.
