How does a moving bicycle remain upright?
Magic, of course!
Sorry, I understand it, but I really couldn’t explain it properly, so I’ll let some of the more physic minded folks explain the principles of gyroscopy to you.
Welcome to SDMB, hope you know what you’re getting into. 
Obviously it doesn’t actually remain upright all by itself. The rider still has to balance it. But what the gyroscopic action of the spinning wheels does for you is to make the bicycle fall over slowly when it starts to fall, giving the rider plenty of time to react.
Physically, without using calculus (notice how when I say that you get the impression that I could use calculus if I wanted to. Hah!), what happens is that the spinning wheels have angular momentum and changing that angular momentum (i.e., tipping the wheel sideways) takes more force than it does for a non-spinning wheel. Subtract that force from the force causing the wheel to tip (gravity) and the net force is smaller.
Now here’s the hard part: according to Newton, acceleration is proportional to force. Less force means less acceleration. Less acceleration means moving (falling) more slowly. Falling more slowly means more time to react! Ta-da! It’s so simple a child could do it.
The physics of bicycles is a weird and wonderful topic. You hear that it’s gyroscopic action and, while that has something to do with it, there’s a lot more. This issue was addressed in a Physics Today article abnout 25 years ago. You might want to check out Jearl D. Walker’s “The Flying Circus of Physics” (the ultimate reference on oddball physics) and look up the references therein.
Speaking from the experience of one who went over the handlebars of my bike way back prehistoric times resulting in a broken clavicle, I dispute that I had plenty of time to react.
Those bike helmet thingies are a good idea.
In the same vein, I’d like to add the following caveat.
This theoretic lab situation obviously does not take the Amsterdam tram rails into account. Plenty of time to react? If you’re lucky, you’re on your feet before the next tram crushes you and your bike 
I notice you said you went ‘over’ your handlebars. In this case the gyroscopic action of your wheels helped you (you’re going in the same direction as the gyroscope’s spin…not perpendicular to it). I discovered the same effect when applying only my front brake. I didn’t break anything myself but I left far too much skin on the road.
I think both of our experiences prove that human stupidity is indeed the greatest force in the universe…FAR outweighing mere Newtonian physics.