Don't kid yourself - nothing is gonna change day to day as far as fanfic is concerned. You'll still have all the same fanfic sites.
If, and it's a whopping big if, Star Trek ever gets included in Amazon's master plan, you'll just end up with three levels of Star Trek author rather than the two you've got now - amateur and professional. A new category slips in between those two that combines a little from column A and a little from column B - an amateur Star Trek writer who gets paid a little for their work.
As for why people will pay for fan-fiction when it's abundantly available online, the answer is easy - the vast majority of fan fiction, whether it's Star Trek or whatever else, is complete shit.
Official Star Trek books are ultimately fanfic like everything else, but fanfic that has been edited multiple times to increase the quality.
Amazon are gonna check the fanfic it puts up to make sure it's legal, family-friendly, legible and has at least some level of literacy to it, again unlike most fanfic that is put onto the internet.
If people who are already writing this stuff can make a quick buck from it then more power to them, cos don't pretend you have any ownership over the stuff now whatever characters you create.
1) Paying for it doesn't mean it isn't going to be shit
2) And Amazon isn't gong to check for anything other than the most basic forms of compliance with formatting and the no slash rules. They're not going to curate the submissions for content quality--just like Kindle, they don't really give a shit one way or the other. Kindle is a platform, not a publisher in the traditional sense, there's no editors or proofreaders to go over your work; no editor to say what is and isn't worth publishing. Freeing and fair in one way, a damn slog through a swamp other ways.
Don't think for a minute that this is going to be much more than a revenue generating scheme version of fanfiction.net at the end of the day.
As for Trek getting into the party: If CBS thinks they can make free money on it, they'll likely do it.