This isn't just an event that happens on a single day, but over a number of days at this time of year.
The reason you see so many is that regional factors affect groups of colonies at the same time, and as nests can contain to 20,000 ants, the swarms seem to reach Biblical porportions and huge clouds of them appear in small areas.
They swarm because they are setting out to try and set up a new nest and they swarm in huge numbers because the attrition rate is so high that perhaps only a few will survive to become new queens and perpetuate the species