It flys in the same way a glider flys. A glider generates lift over the wings, but the lift comes from the airflow, and you get the airflow by trading potential energy for kinetic energy. Thus the glider is always descending through the airmass, albeit at a very shallow glide angle. But the airmass itself isn't stationary - the sun heats the ground unevenly, so some air above it is warmer and rises, and some parts are colder and sink, with winds circulating between them. A glider can stay airborne for hours if the pilot is good at finding rising air, and circling within it. So long as the air is rising faster than the glider is descending within the airmass, the glider will gain altitude.
The dandelion is doing the same thing. It's creating lift by descending slowly within the airmass, though the process of creating lift is via the vortex ring, rather than airflow over wings. Its aim is to fall very very slowly. Then it will often be falling slower than the airmass is rising, and it will be carried up and far away. Of course there's no pilot, so it can't actively seek out rising air, but there are an awful lot of dandelion seeds, and only a few need to get lucky to spread a long way.
Some seeds have a small leaf attached, making them spin as they fall and slowing their descent for the same purpose. It's not about flying, just about prolonging the time spent in a moving body of air.
I imagine they get blown with the wind, start falling, which triggers the formation of the vortex - and that vortex is somewhat self-sustaining, almost completely arresting the fall.
I do have one question though, how is the air flowing up through? It the seed not falling for that to happen?