Apple owned the tablet market when Amazon entered the fray. By last Christmas, that market was pretty competitive with Apple taking about a third...
Apple shipped 26m iPads in 4q2013. The total of all dedicated streamers sold EVER is less than that...
While serious gamers invest in the top platforms (PC/Xbox/PSx/Wii), a LOT of people are playing games on a tablet instead or as well...
In fact, among young kids, games are the main use of a tablet. And, btw, consoles are a niche comparable to streamers...
One more graph: what people are streaming...
While none of this was carefully researched, it supports my premise, so I went with it. My premise is simply that a lot of people are doing the things that the Fire TV does, but a lot more people are not. While no one is going to trade in a PS4 or an Apple TV for a Fire TV, people looking for a new device or a streamer for their 'other' televisions may well find that FTV does what they want for a reasonable price. When parents are looking for a streamer for thier youngsters, they will find that FTV supports familiar games, familiar services, and familiar controllers. They will be comfortable with the seller (Amazon) and will like the parental controls and FreeTime -- even if they never get around to using it.
One more thing: anyone remember living books? Each came with a paperbook booklet. You could read that book, read the book on your PC, have the PC read the book to you (highlighting the text as it read), and even 'play' on each page of the interactive book. This would be a lot of fun on a 60" plasma.