That's because, unlike HBO's other apps, including HBO Go and HBO Now, Apple's version of HBO inside Apple TV Channels lets you save episodes to watch offline when you don't have an internet connection.
Yes, you can also watch "Game of Thrones" and other HBO shows offline if you have a digital DVR subscription with your cable provider, which might allow you to record the show and watch it at a separate time. Or you can do so if you've purchased all 7 prior digital seasons of HBO. But that can be messy, and you can't always just go back and pick an episode unless you've recorded all of them and also pay for cable.
So, if you just want to subscribe to HBO and download any episode you want, whenever you want to, Apple Channels is your best bet. And you'll want to cancel your existing HBO subscription if you have one, since you're basically signing up all over again, just through Apple this time.
Note: You won't be able to access this HBO subscription from anywhere outside of the Apple TV app, which means this subscription won't work on HBO apps you might have installed on a Roku or Fire TV until Apple rolls out its Apple TV app for those platforms.
If that's all agreeable, and it might be if you only want HBO on your iPhone, iPad and Apple TV at home, then let's move on.
That's it. Now you can watch the episodes you want by opening the Apple TV app on whatever device you've downloaded the show to. If "Game of Thrones" or HBO aren't your cup of tea, you can also do this with Apple's other premium partners such as Starz and Showtime.