I just started a martial arts class in a studio that is primarily a Karate studio for kids. I'm curious about this.

There's a sort of carpeted store/spectator area and a training area with mats on the floor. Everyone always bows before stepping onto the mats. It's seems to me that this is a tradition setup by the studio's owner with its roots in Asian martial arts tradition. They said it's meant to show respect for where you train. But I want to learn about the beginning of this tradition, if possible.

How did this idea of showing reverence and respect for your training area develop? Is there some sort of pedagogical reason (e.g. to put you in the correct state of mind) that this bowing tradition has persisted over the years?