This gets into the "Control Theory" branch of engineering, but here's the simple form.
Systems that are stable are systems where perturbations are reduced, and you return to a normal or set state after a time.
Unstable systems are where, when perturbed, the system goes out of control.
Think of a marble on top of a dome, versus one in a bowl. You can maybe balance a marble on the top of a dome, but if you shake it at all, it will start rolling off the edge - and roll away faster and faster. Alternatively, a marble in a bowl, if you bump it, will sway back and forth a bit, and settle back into the very center.
More complex systems often have several unstable components, with correcting components such that, within certain parameters, it is a stable system. Let me explain with the bike:
A bicycle is a great, stable system. That's why they're so easy to use. There are two main componenets to bicycle balance.
The first component is gravity, and it is unstable; when you lean left or right, you move your center of mass away from right-above the pivot. And the further away you are, the more significant that torque will be from your weight. It's very unstable. If your bike is stationary, if you get more than a few inches or so off-center, it is very difficult to recover and you will fall over.
The second component is the correcting, stabilizing component. When you are moving, and you lean to the left, you start turning left. This means you are now pointed a few degrees to the left - but you still have forward momentum. This forward momentum tries to flip you over the opposite direction, pushing you back up-right (pun intended).
In effect, whenever the bike leans to one side or another, a force is generated that tries to push the bike back to being straight-up. But this corrective force depends on your current velocity. If you're not moving, there is no corrective force, and you just tip over. If you're going slow, its still pretty weak and can't be relied upon. This is why starting a bike is the most difficult thing, and why children have trouble learning initially. Once you get them moving, the bike pretty much balances itself as long as the rider keeps it moving and doesn't deliberately tip it over.
So a bike tries to tip over just because gravity tries to pull it over. And it doesn't tip over because forward momentum rotates you back up, when you're going fast enough.
What happens if you go faster?
Well, the correcting-force will be stronger. So when you just slightly rotate to the left, you'll receive a very large force rotating you to your right. That force might be so large, that it pushes you over to your right. Now you begin turning to the right. The catch is, that that force is still pushing you over to the right because you're still pointed to the left relative to your initial forward momentum.
After a short delay, you'll turn enough to the right that you're now pointing right, relative to that forward momentum, and now the corrective force is trying to rotate you back left. This delay is the catch. If the delay is short, you'll go back and forth like this. This is the wobble.
If the delay is longer then you'll over-correct too quickly, before you turn enough to correct the over-correction. The faster you're going, the longer this delay. During this delay, both gravity and your correcting force are conspiring to tip you in the same direction. Because you have to be at high speeds for this to happen in the first place, the correcting-force is pretty strong, and very difficult for you to counteract by shifting your weight.
At this point, any small mistake you make will result in those forces winning, and you having a nasty crash.
I hope you were able to follow that. I don't feel I was too clear.
TL:DR
Bike tries to fall over, but leaning to one side while moving forward makes you turn that direction. Pointing in a direction will cause your forward momentum to tip you in the other direction, pushing you back up-right. When stopped or at slow speeds, this corrective force is weak, and balancing is difficult. At normal speeds, this force is sufficiently strong to push you back up-right, and you can balance very easily. Basically the bike is balancing for you. At fast speeds, this corrective force over-corrects, not just pushing you upright, but pushing you over to lean to the other side. This will repeat and make you wobble.
Once you start going too fast, you leave the nice range of stability where the bike balances for you, and you have to balance yourself. The bike becomes more unstable the faster you go, so you have to work harder and be more precise to keep it upright, and it becomes more and more difficult until you can't maintain it anymore and fall over.