It means that there were a great many people in the square, and a lot of things happening there. The verbs 'swarming' and 'buzzing' suggest the movement and sound of insects. Bees and wasps, for example, 'swarm' and 'buzz'. These images reinforce the idea of a a great multitude of people moving around.

The metaphor is that there were so many people there that they were like insects, swarming and buzzing. So lot's of people, moving around, doing things, like watching a swarm of bees or ants.