This bit is easy to understand in an intuitive way if not mathematically. Spin a coin on its axis and as it slows down it will flatten out so that the contact point between the edge of the coin and the table ‘moves’ around the coin. Roll an old tyre along the ground and see what happens as it slows and starts to fall to one side. The tyre is turning around its own curvature. Push your uni slowly and turn it and see what happens.
On a curve you are in a similar position to a body in orbit. Your momentum from the straight will tend to push you away from the centre of the corner (keep you going straight; make the uni stand up), and your lean will fight against that. On a smooth turn, the forces are balanced for longer; on a sharp turn, there are sudden imbalances and corrections.
But how do you start the turn? That was a good question. Intuitively, you’d think Newton’s laws would say that you can’t move your centre of mass sideways without something to push against. That is right. However, you can move part of your mass, by moving it agains the rest of your mass. So, you can start the uni turning by twisting your lower body, or by pushing the pedal in a certain way, by using your main body mass as something to react against.
You only need to start a slight lean of the wheel which then rolls out from under you a bit, and the centre of support (tyre contact patch) is no longer under the centre of mass. Then gravity starts to contribute (sometimes more than we would choose ) and you need to accelerate to counteract the fall. You accelerate out of the turn, or steer the wheel back under your mass, whichever way you look at it.
So, to start the turn, part of you pushes against another part of you. Legs go one way, body goes slightly the other. To continue the turn, the natural curve of the wheel does the job. To end the turn, either countersteer or accelerate.
In real life, of course, it either works or it doesn’t.
As for turning smoothly… an excellent objective. For muni or road riding, you often need to follow a very specific path. When playing or performing, sudden sharp turns are more fun.