Based on this paper (thanks to Oliver_C for posting it) you can quite clearly see that their behaviour is understood by physicists (they wrote a paper on it!)
To find the essence of bicycle self balance we looked at simpler and simpler dynamical models until we found a minimal two-mass-skate (TMS) bicycle that theory told us should be self-stable. This bicycle has no gyroscopic effect and no trail. We built a bicycle (of sorts) based on the theory to prove the point.
This shows that theory predicts their simple bike to be stable without gyroscopic or trail effects.
Gyroscopic forces and trail effects DO help bicycles remain stable, however. This paper just demonstrates that other effects are important too, and that bikes can be stable without gyroscopic forces or trail.
Why can a bicycle balance itself? One necessary condition for bicycle self stability is (once we define the words carefully) that such a bicycle turns into a fall.
This situation seems similar to that of aircraft, where the Bernoulli effect is often cited as being the cause of lift, while planes can fly without it (for example when they fly upside down, which some planes can do.) The gyroscopic effect certainly helps balance a bike, but isn't necessary, much like the Bernoulli effect helps give an aircraft lift but isn't necessary.