Can anyone explain to me why 準備が出来ている means "It's ready?" I don't think I know exactly what 出来ます means, as we initially learned it as "able to"- but I've seen other uses like in saying "I have/got a girlfriend," or "There is a library." (At least, I think the latter, like 図書館が出来た." In any event, I know 準備 is preparation, so the real question is what does 出来ている mean, and why does it mean that?
There are two uses of できる:
As a potential form for する, meaning "can do". Like the potential forms of other verbs, it's stative and generally can't appear in the 〜ている form. Compare the ungrammatical English "I am being able to do 〜".
As a verb in its own right, できる means "come into existence, complete, accomplish, perform well", which can have a variety of meanings in different contexts. It can be used for puddles forming, or for making friends, or for being pregnant, or for doing well on a test.
In this meaning できる isn't a stative verb. It's a punctual verb, and with 〜ている it has a resultative meaning:
Preparations are complete
(Punctual verbs are also called achievement verbs.)
出来る (usually written in kana) can take on several different meanings (see e.g. http://tangorin.com/general/出来る for the entire list). Here it means that the preparations "became finished".
In the sentence "図書館ができた", it means that a library was built.