There's also a sort of "common sense" explanation relying on really basic physics:
Let's say you want to turn left. We know from experience that you will be leaning left when you turn: meaning, your weight is going to be on the left hand side of your wheels. (The common-sense explanation works really well because I'm not going to explain why you need to lean left to steer left, I'm just going to take it as a given.)
The basic-physics question is: if you're starting totally vertical, how the heck do you get your body mass to be on the left hand side of the wheels in the first place?
You can get a little bit of effect, of course, by simply leaning over left: but that leans the bicycle to the right because you can only lean over one way by pushing your bicycle the other way... and when you do this the bicycle briefly opposes you in the way it's leaning before your gravity can win out. That's less good.
So what your brain has actually learned to reflexively do is, to send your wheels going off to the right while your body continues in a straight line, hence your mass is now over the left-hand side of your bicycle and there is no competition. You then steer into the turn and complete it normally.
http://socrates.berkeley.edu/%7Efajans/Teaching/bicycles.htm...
with plots of torque-vs-time and analysis here:
http://socrates.berkeley.edu/~fajans/pub/pdffiles/SteerBikeA...
The author spends most of his time at CERN, but got interested in bike physics for a couple of years.