When you prance, you swagger around with exaggerated, proud movements. If you've seen many rock concerts, you've seen people prance around onstage.
When people prance, they're showing off, strutting for the benefit of their audience. Horses prance too, with a gait that's also a little artificial, using high, springy steps. You might prance a little when you run up on stage to accept an award, and a show horse might be equally inclined to prance as it moves past the judges. The horse gait definition is the original meaning of prance, and it probably comes from the Middle English pranken, "to show off."