If you've read Christopher McDougall's book "Born To Run" (if not, I strongly recommend that you do), you will understand that humans are literally made to run. That is, our ancestor ran a lot in order to survive (be it after food, away from predators, and as a way to cover distances for various reasons). And, guess, what?! They did not possess the latest shoe designs with air pockets, straps, ankle support, inserts, or other bells and whistles. Their footwear, if they possessed any, was nothing more than a piece of leather which protected the foot from scrapes, blisters or cuts. As a result of this lifestyle, their feet (and, as a result, OUR feet) were shaped to resemble carriage springs (upside-down "U" shape). Accordingly, when people run barefoot, the forefoot tends to strike the ground first which then automatically propels the person to "spring" forward again.
However, due to technological developments (and our sedentary lifestyle) our feet have not only been weakened but our stride has shifted from toe-to-toe running to heel-to-toe rolling. The problem is that spending too much time on your heels is not good for tennis. Tennis players need to be EXPLOSIVE, they need to SPRING to and away from the ball. That's why coaches emphasize "playing" or "moving" on your toes. This is that same movements that boxers use in the ring and, consequently, why boxers AND tennis players need to spend a lot of time jumping rope (in order to strengthen the muscles in their feet and ankles - which allow them to bounce around the court).
A good way for players to understand what "moving on the toes" means, is by having them take their shoes off for a few drills (they can keep the socks on). These, obviously, don't need to be very difficult drills. Just some basic side to sides, inside-outs, up-and-backs, returns, etc. By being barefoot, the player will NATURALLY grasp the concept of shifting the weight forward onto the balls of their feet and keeping the feet "under their bodies" (as opposed to sloppily getting off-balance). It will be very uncomfortable for the player to lean their weight back on their heels and, God forbid, actually land on their heels. The players will try to avoid this pain at all costs and move the way nature and evolution intended us to move - on our front-foot. After the player feels that she has learned the concept, re-do the same drills and have the player explain the difference. This is a very fun way for players to learn proper footwork and balance.