I figured it out all on my own – I am good at this physics stuff 🙂 If you hold the bike wheel still and tilt it so the part moves left at the top part moves right. Take a particular chunk of mass at the bottom, it has momentum to the left. If the wheel is rotating, however, that same chunk still has momentum to the left when it’s at the top, which means it’s now opposing your tilt. So all the force you put into tilting it ends up opposing the tilting force a short time later.
The faster you spin it, the shorter that time lag is so the better it works. This also explains why you can get it to oscillate if you wiggle the wheel side to side at the right frequency. That frequency matches the angular momentum of the wheel. Hence speed wobble on motorbikes, I suspect.