In simple terms, your steering input has a more immediate effect on the relationships that apply balancing forces to the bike if you are moving faster.
The general solution is actually a hard one to eli5! It’s complex area of dynamics, partially with regard to self-stabilisation of a bicycle in motion. If you are really interested in investing effort in understanding it [Wikipedia](https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics?wprov=sfti1) is as a good a starting point as always.
Without considering self stabilisation, and answering the very specific question of why it’s easier to balance at speed, it’s largely because any given input (with the corresponding change to geometry of the system) will result in a more rapid displacement of the center of mass versus the points of contact with the ground around which the bike ‘pivots’ when the longitudinal displacement of the system per unit time is higher. This means you can more quickly apply a correcting moment to the bike with your inputs when it starts to topple.
(Some of the responses in this thread are bananas !)
Angular momentum – that’s not ELI5, but it’s a major reason for stability in rotating objects. The simple way to think of it is like this. When you spin a top, it’s axis is aligned vertically. Because it has mass, and is spinning, that axis wants to stay vertically aligned. Some concept with bicycle wheels except you have two of them and it wants the axis to stay aligned horizontally.
You are constantly falling. The speeds let’s you not fall because of steering. If you don’t have one imagine it in your head. While stationary tilt a bicycle to the left close to the ground. Turn the bicycle to the left with a sharp turn and push it forward. The bicycle will pick it self up as you push. Now while in motion instead of pushing, you have momentum doing it for you.
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