It's all about friction. Ever lean into a turn on a sand-covered road. You lose grip and BAM! Friction is what keeps you in equilibrium. The tires are trying to get back under the center of mass (aka center of gravity) and at the same time, the bike is trying to fall over due to gravity pulling it toward the Earth. The two actions work against each other and the bike is balanced while being leaned over.

You ever balance something that was tall? a big stack of cardboard boxes or a broom sticking straight up in the air? If the broom starts to lean to the left, you quickly move the bottom to left to "get under" it. If it goes to the right, you move the bottom to the right.

Imagine if the bike was not moving. You lean the bike over and let it go. It falls over.

Now, if you had a way to pull the tires quickly in the direction of the lean, you could balance the bike back upright.

Then, if you were able to move the tires quickly in the direction of the lean, but only quickly enough to keep the bike from falling any further than it already had fallen (let's say 20 degrees off the vertical), then the bike would stay leaned over because the force you are applying to the tires is exactly equal to the force of gravity on the Center of Mass.