I just tried it yesterday at my friends place, played for a few hours. However, I'm confused about the beginning; if it's explained later in the game, I would rather not know, but if I simply misunderstood something (i have played Arkham Asylum), I'd like to know.

Spoiler contains the first 25 minutes of the game more or less.

Did I get it all wrong, is there an explanation later, or is it simply not explained?

Theres a comic that goes deeper into what happened between the games, but as you progress further you'll start filling in the blanks yourself (i hope)

I'm fairly certain that during the opening sequence, Vicki Vale references Arkham City as a sectioned-off part of Gotham that has been converted into a giant prison and that Bruce Wayne is protesting its existence. Also every prison has medical staff.

As you finish riddles you'll unlock "Arkham City Stories", which are little vignettes that explain why things are the way they are. There's also story stuff interspersed that'll give you a pretty good idea of why things are the way they are.

Comic Vine has got your back:

http://www.comicvine.com/news/what-you-need-to-know-before-playing-batman-arkham-city/143737/

Well everyone seems to got you covered! The comicvine link is great if you didn't read the comics to introduce you in.