OK, they are also stopping Chromecast sales too - but its the Amazon/Apple relationship I find interesting. I think Amazon see themselves as a kind of "Google in a parallel dimension" kind of player - everything Google does is about a free and open internet (free and open to Google). But Amazon would quite happily have a bunch of content (films, books, TV programmes etc. etc.) paywalled off from the rest of the internet.
Apple, on the other hand, seems like more of an Amazon nemesis. Like Google, Amazon could be enormously threatened by an Apple monopoly. In fact, I would guess that the whole point of the Fire phone was ultimately the same as Android - but aiming to fill a gap where Android's free/open strategy was threatening to open up (ie. at the high end where Apple dominates.) It was a pretty clear failure, so Amazon have abandoned the strategy. (And lets face it, the damage was done back when they were fighting over having a "buy" button in the Kindle app without letting Apple set the rules over how in-app purchases should work.)
But the Apple TV thing seems a bit weird. When Apple announced the new Apple TV would be a more open platform with an app store etc., the obvious thing for Amazon to do would be to build an Amazon Prime app to let their subscribers watch their content on their Apple TV boxes. Instead of saying "no, we aren't doing that" — or maybe rather, instead of just saying no, they are saying that not only aren't they playing with Apple's boxes, and they aren't going to be stocking and selling Apple's boxes, and they aren't going to let anyone sell Apple's boxes in their marketplace.
Seems a surprisingly bold statement to me. (The NYT says it mirrors Apple's refusal to allow Netflix onto the Apple TV platform - which must have been from before I can remember. Must have been before their UK launch.)
But the bottom line of all this is that the future isn't going to be one smart box connected to one dumb TV. Its going to be a bunch of things connected to some sort of dream box that makes it all manageable. But still connected to a dumb TV...