As much as it sucks for the author, if you're providing a service or app on top of another service (like providing a better way to browse a website's content or API) you're really at the mercy of that service provider, especially if you've registered to use their API. If they want to squash your service/app, they can either just revoke your API access, or just find some violation of their TOS, or just create one with a TOS update. Then they can report you to the respective app store (App Store, Google Play Store) and most likely get it removed. Without looking at the Youtube TOS, I wouldn't be surprised if there was a section that explicitly disallows paid apps besides Youtube's, probably to prevent competitors for their Youtube Red app.

Although I wouldn't have bought it since the plain Youtube app is fine for me, it looks like he put a lot of work and soul into the app, and I feel for him. I also appreciate the level headedness of the statement, explaining the situation without much vitriol, although I'm surprised he doesn't mention anything about one of the issues Google likely had, which was ad-blocking (see edit below). Good luck to him on his next project.

Edit: It looks like the author of the app likely got in a lot of trouble with Google for not having ads when playing videos, on top of the other issues, a commenter below points out. I think it's a little disingenuous of the author to not put this in his statement as I'm sure Google wasn't happy about it (and probably said so in the emails) and it's likely another violation of the TOS (and I'm surprised they haven't pulled/didn't pull API access). What I said still stands, but he should be more direct about the issues that Google had.