I’m having the exact same problem as the original post. The first image below shows the same movie as the original post, and you can see that it is almost identical. It is happening on a wide range of movies. I cannot pinpoint an exact date when we first noticed this, but I believe it was in the last week or two. Random sampling seems to point to it only happening on movies with a large number of studio credits, or possibly a large # of credits in general. In the case of Wonder Woman (the 2nd image below), the layout problem is actually interfering with the ability to select buttons (if you swipe to the right from Play, selection will actually drop down to the behind the scenes video below).
I have reinstalled the Apple TV app as well as the Plex Media Server, but neither has helped.
Platform: OS X 10.11.6
Plex App: 1.24.1 (8229)