What is a CustomPainter class, and how does its paint method let you actually issue the specific drawing instructions rendered by a CustomPaint widget?

A CustomPainter is an abstract class that you extend to define your own actual drawing logic, and its required paint method receives a Canvas object representing the drawing surface along with the available Size to draw within, letting you call various drawing methods on that canvas, such as drawLine, drawCircle, or drawPath, to actually render your desired custom graphics, with the CustomPaint widget itself responsible for calling this paint method at the appropriate time whenever the widget needs to be rendered or redrawn.

class MyCirclePainter extends CustomPainter {
  @override
  void paint(Canvas canvas, Size size) {
    final paint = Paint()..color = Colors.blue;
    canvas.drawCircle(size.center(Offset.zero), 50, paint);
  }

  @override
  bool shouldRepaint(covariant CustomPainter oldDelegate) => false;
}