I’m not sure about describing formation, but I think it’s quite intuitive to understand how a vortex is maintained, and why the center is at low pressure. Assume that a vortex has already formed, and consider the movement of molecules at an instant in time. Considered instantaneously, all the rotating molecules are moving tangentially, i.e. in straight lines parallel to the circumference of the eye. With no other force applied, a molecule will tend to keep moving in a straight line. But a continued straight line would takes these tangentially moving molecules away from the center. Fewer molecules at the center means reduced pressure at the center, creating a pressure gradient that will tend to bend the molecules around. So there’s a dynamic equilibrium condition where molecules are all moving tangentially, and there’s enough of a pressure gradient (low pressure in the middle) to bend the molecules around into a perfectly circular motion.
Given that there are three dimensions with ground below, it’s also easy to see intuitively that there’s a tendency for this low pressure in the center that’s created by the circulation to pull air down from above. So the downward movement of air is an effect, not a cause.