In modern times, this expression is taken to mean that a healthy diet is a basic means of maintaining your health, and as such, you won't require the services of a doctor. This clearly demonstrates our faith in the abilities of modern medical science. In previous years however, medical knowledge was much less advanced, and the doctor was a figure of fear.

In medieval times, a visit from the doctor would be likely to result in great pain, and often the treatment would be ineffective in any case. Treatments would include such as bleeding, noxious chemicals, and even amputations. The doctor would also charge extortionate amounts for his services (thus also creating the notion of a large charge costing an arm and a leg.)

Understandably, people would come to dread a visit from the doctor, and wouldn't dream of actually calling one out. The doctors would thus be unable to make a living, and so began to wander the streets looking for potential victims, or patients as they later became known. This obviously worried the general public, but then someone realized that the doctors weren't treating people out of any desire to improve the well being of the population, but instead purely to make money. This, like all the best inventions, seems so obvious in retrospect, but took a genius to realize.

Once it had been discovered, a simple solution immediately presented itself - if you fed the doctor anyway, he would have no need to wander around generating income. The simple practice of giving the doctor food on a regular basis meant that he would leave people alone. Giving the doctor an apple a day (or any other food come to that) would keep him away.