So hold on. I have DirecTV at home. I own an Apple TV and have Food Network and other apps installed to watch shows on my Apple TV. But if I use the TV App on my iPhone, Food Network comes up, I tap on a show, tap Play....shouldn't it play in the Apple TV app now...? I still need to download the Food Network app on my iPhone?
Yes. Apple TV Channels are for subscriptions through the Apple TV app. It's essentially a cord cutter app. If you no longer subscribe to TV through a cable provider, you can subscribe to individual channels from within the Apple TV app.