What is the purpose behind "wax on, wax off" in Karate Kid?

Miyagi becomes Daniel's teacher and, slowly, a surrogate father figure. He begins Daniel's training by having him perform laborious chores such as waxing cars, sanding a wooden floor, refinishing a fence, and painting Miyagi's house. Each chore is accompanied with a specific movement, such as clockwise/counter-clockwise hand motions.

Daniel fails to see any connection to his training from these hard chores and eventually feels frustrated, believing he has learned nothing of karate. When he expresses his frustration, Miyagi reveals that Daniel has been learning defensive blocks through muscle memory learned by performing the chores.

Mr. Miagi's shows him all the specific task he had been doing and how they transferred into karate. At the end of the transfer of learning phase of his training Daniel is amazed at what he had "learned" when he thought he was just doing everyday tasks.

An odd form of training passed off by an unorthodox master on a skeptical student. Sometimes comes disguised as a set of chores, but just as often is a general exercise that promotes a valuable physical or mental attribute in a strange way. Always dismissed as a waste of time early on, and appreciated later. Often this also serves as a lesson to the skeptical student to trust the master and do all the crazy things the master asks without questioning, by demonstrating that the master really knows what he's doing and is in fact effectively teaching the student.

It can be considered also a life lesson, and you can read more about it in this nice blog post over on a China travel site. It's like your childhood. When you're young you always complain with your parents for being so severe, but only when you're an adult you fully understand that your parents acted like that because they wanted you to become an educated person.

Originally it's from the movie The Karate Kid(1984) starring Ralph Macchio as Daniel and Pat Morita as Mr. Miyagi and the phrase has been adapted to the remake too.

Mr. Miyagi was supposed to be teaching Daniel karate, but all summer the karate master had the kid just painting stuff and waxing his car. However, the painting and waxing techniques were secretly teaching his muscles karate moves, and when the karate lessons actually began, he found it relatively easy to learn. This technique is known as the Kagite uke or circle block. It is a very effective block that provides a good lead in for follow on techniques.

“There are some skills you can learn on your own, and some you can try to learn, but if you intend to take the journey of mastery, the best thing you can do is to arrange for first-rate instruction. For mastering most skills, there’s nothing better than being in the hands of a master teacher, either one-to-one or in a small group.”

I think,the purpose of "wax on, wax off" would be not to to teach him the value of hard work, but to teach him the basic karate movement and make him understand mastery in karate is a long-term journey.

I don't think the purpose was simply to learn the muscle movements for defense. With the struggle of teaching, I think Myagi was also teaching the boy to trust him. Trust isn't something Daniel was very familiar with, and developing a trusting relationship with Myagi, as we see later, leads to better things.

The remake actually provides some clarity on the matter. In either Karate or Kung Fu, there is a recurring theme of "excellence in everything you do, no matter how mundane". The underlying thread in this concept is meditation - a truly enlightened mind is meditating all of the time. Martial artists, seeing their art as a meditation, find practice in everything they do. This is the deeper significance to disguising technique as chores.