One of the more puzzling aspects of Apple TV+ is the occasional, fleeting appearance of the picture-in-picture icon, whenever you touch (but not press) the Apple TV remote.
To see this yourself, fire up your Apple TV, then switch over to the TV app and launch one of the Apple TV+ shows (the first few episodes are free, require no signup). As a show is playing, touch the touchpad on the Apple TV remote. A PiP icon will appear. See if you can figure out how to turn on picture-in-picture mode.
Turns out, this is a tvOS 13 feature, specific to the TV app, and not specific to Apple TV+. If you tend to use individual apps like Netflix or Hulu, rather than the TV app, you might never have encountered the PiP icon. But since Apple TV+ lives in the TV app, you’ll definitely encounter the PiP icon once you move to Apple TV+.
As to the specifics of using PiP in tvOS 13, TidBITS posted an article when tvOS 13 dropped last month that lays it all out.
The key is to swipe up when you see the icon, then click the touchpad to bring PiP to life. Read the article for all the mechanics of swapping PiP window positions, swapping video between full screen and PiP, all worth knowing.