Almost as if this is just yet another get rich quick-scheme that only works for a select few, but gets hyped due to the success stories spreading easily? Who'd've thunk it?
There's no real "oragnization" behind. The various coins are regulated by whatever body "owns" them (and thus has power to change the hashing algorithms, change how mining difficulty rises, etc), but nobody has control over the output beyond "anyone linked to our system gets a reward based on the amount of work they feed into the system". That's where the illusion of this being democratic comes from, I guess. They can still change the algorithms dramatically or increase mining difficulty - which has been done previously (scheduled and notified well beforehand IIRC) to avoid inflation and so on. The whole "how do they know when I am supposed to get something back" is from what I understand the new thing about blockchain, that it creates an unreproducible but traceable and unique proof of work. So it's built into the system, so to speak. There's no model answer, as the entire point is that the only way to get a result is to do the work, and each proof of work is unique. I have no idea how it really works, but then I don't see enough of a point to this whole thing to care enough to look into it either.
It's far easier understanding how this has value at all, and isn't just seen as pissing into the wind: because as with any financial investment target, its value is fundamentally abstracted, and solely defined by "market trust". I.e. if the gamblers on Wall Street feel like it's still worth something, they won't start selling, so it keeps its value, and if they feel its worth buying, the value will rise. The obvious downside to this is that investors are an extremely irrational and unpredictable bunch, and are prone to mass panics, so things tend to come crashing down at some point. Of course if lots of trading is done by people with a lot of money bound up in this and no real way out, such as organized crime, then that will prevent most catastrophic crashes simply because they won't join in on the panic selling.
So there are no organizations behind it, but once people started seeing this as something they could gamble on to make money, a lot of big players suddenly put a lot of money into it. So the system is held up by a lot of big players, though none of them have any control of how the system works beyond trade and value.