Amazon has a great thing with their Amazon Prime service, offering up plenty of free viewing of loads of popular TV shows from various networks through the Instant Video section, but one thing that Amazon lacks with Prime and the instant video portion of the service is the capability to stream videos to the Chromecast through a compatible Android app. Yes, Amazon has their own competition for the Chromecast with the Fire TV Stick, which has many of the same available streaming video options as the Chromecast, plus more games. What they don't have though is the majority of usership. That title goes to the Chromecast as it is still the number one selling streaming device on Amazon.
This is where Amazon could capitalize and potentially reel in more subscribers to Prime. They already have an estimated 10 million Prime subscribers, but chances are many of those users don't own one of the Amazon Fire TV streaming devices. Chances are that many of them don't own a Chromecast either. I'd be willing to bet though that more Prime users have a Chromecast than a Fire TV product, and opening up their instant video service to Chromecast users officially would not only give existing subscribers more choice, but for those who already own a Chromecast and love it but don't yet subscribe to Prime, the appeal of being able to get their hands on and stream some of the video content you can find in Prime could be too good of a deal to pass up. If it existed.
You might think that they could be hurting their own overall sales of the Fire TV Stick if they open up Instant Video to a competing device, and that may be true to some degree, but the Fire TV Stick does offer some content that isn't widely available via Chromecast, like games for example. Not to mention a handful of streaming video options that Chromecast(at least at this point in time)isn't compatible with. Those that wanted those extra features could still opt for the Fire TV stick instead of a Chromecast. While the Fire TV stick is great, it still is just another way for Amazon to grab people's attention and focus towards a Prime subscription, and if that's a strong goal for Amazon, why not potentially expand their reach by offering the streaming service to the community of users on the best selling streaming device out there?