If you read the actual paper referenced, we know some of how it works, but not all.
Two of the proposed theories (that it has to do with gyroscopic effects, and that it has to do with trail), have been disproven by creating a self-stable bike with no gyroscopic effects and (slightly) negative trail. The paper introduces one additional factor, the difference in center of mass between the steering assembly and the rigid body of the bike; the steering assembly having a lower center of mass causes it to fall faster, providing the necessary corrective steering to achieve self-stability.
So, there are several factors we know about, which can increase stability. We know how to locally optimize stability for certain designs. But we don't yet have a full set of necessary and sufficient conditions for a bike to be self-stable. We haven't even proven, analytically, the intuitive notion that a bicycle must lean toward a fall, though given our intuition it is believed to be true.
Here are the two necessary conditions that the paper provides:
> To hold a self-stable bicycle in a right steady turn requires a left torque on the handlebars. Equivalently, if the hands are suddenly released from holding a self-stable bicycle in a steady turn to the right, the immediate first motion of the handlebars will be a turn further to the right. This is a rigorous version of the more general, as-yet-unproved claim that a stable bicycle must turn toward a fall.
> Another simple necessary condition for self-stability is that at least one factor coupling lean to steer must be present [at least one of Mδϕ, Cδϕ, or Kδϕ must be nonzero (SOM text S3)]. These coupling terms arise from combinations of trail, spin momentum, steer axis tilt, and center of mass locations and products of inertia of the front and rear assemblies.
That's what is meant when people say "we don't know how bicycles work." We do know some of how they work; we know that the designs that we create steer into a fall, and do so in such a way that damps the wobbles and eventually goes straight again. And we do know some necessary conditions for a self-stable bicycle, like a requirement that something that couples lean and steering must be present; but we don't know if the steering into the fall is absolutely necessary, or if you could build a bike that managed to achieve self-stability without it.
So, I would say that a more accurate summary is "we know how current bicycle designs work, but we don't know exactly what aspects of them are necessary, or how to completely characterize the sets of designs that work or don't work." But that's a bit more of a mouthful than "we don't know how bicycles work", so that's what gets repeated.