I love this description of quantum. The paper Aaronson links to, "Quantum Theory from Five Reasonable Axioms" (http://arxiv.org/abs/quant-ph/0101012), is also a great read.

The "probability in L^2" cleared up a lot of confusion for me, although I still have a very poor intuition for what the Born probabilities are probabilities of. If you believe the MWI story, it seems like it's the probability you will "find yourself in the universe where this outcome happened" but even that sounds odd to me.