I'm surprised how much I'm seeing gyro brought up in these comments. It's a factor, but it's practically negligible. It's all in the steering. Start to tip right, and you'll subconsciously steer slightly to the right to correct your balance. Try to ride as slow as you can and you'll find yourself doing these corrections much more frantically and dramatically. The reason for that is because it takes longer for the wheel to roll under your center gravity and "catch" you when you're going slowly so you have to turn in quicker to maintain balance.
Notice that on almost every bike you see, the front axle on the bike is slightly ahead of the neck's axis of rotation. That offset does two things: 1. It stabilizes the steering so that the bike will tend to steer straight and 2. (more important to my point) It makes the balance-correcting effect of steering more immediate and dramatic, making it much easier to ride at slower speeds.
As a counter argument showing why gyro is barely a factor, these exist:
Edit: if you're not seeing the image like I'm not, Google "ski bike".
Snowboards definitely have steering, you twist the board and shift your weight to manipulate how the edges contact the snow, it's just not quiet as explicit as a bikes front wheel. But whether it's a bike, a board, or literally any moving thing on land, the steering happens because you applied a lateral force to the ground and an equal and opposite force was applied back to you.
The snowboard uses different methods of applying that force, but other than that it's the same concept as described in my first comment: Greater speed allows more subtle corrections to take effect more quickly.
Now the snowboard does have a wider contact area with the ground, but that really only helps you on flat ground at very low speed, or standstill. Advanced boarders will carve transitioning from edge to edge most of the time.