The spinning wheels have angular momentum. It's a vector, so it tends to keep the same orientation as well as speed in the absence of an outside force. This means it's harder to make the wheels change their angle with the ground when they're spinning faster. They act like gyroscopes and keep the bike stable.
To see this, hold a bicycle wheel vertically with one hand on the axle on each side so it can spin. Get someone to spin the wheel. Now try to twist the wheel to be horizontal. It will feel like it's fighting you and trying to stay vertical. The angular momentum "wants" to stay the same.
The faster you go the more stable you are.
