What is the purpose of oiling a wooden chopping board?

The oil is not actually doing anything to protect the wood - the idea that it does so is a very widespread myth. This is the conclusion of the guy who wrote the book on wood finishing, and I can confirm it from personal experience. You can continue using it for cosmetic reasons, I am personally partial to the look given by flax oil - although there are people who prefer the opposite, because cuts are much less noticeable on unoiled wood.

To get real protection on an wooden item, you have to apply a layer that actually seals the wood. The substances which do this well are shellac, nitro lacquers, acrylic finishes and alkydic finishes.

While you could use a safe-in-trace-amounts finish on a chopping board, the mechanical action of a knife will soon damage such a layer to a point where it becomes useless. Most people are just not interested in refinishing their chopping board a couple of times per week.

As for the source of the myth: Long before good finishes were discovered, people tried protecting wood. Since it was known that wood is most damaged by moisture, they used all kinds of water-repelling substances, such as different oils and waxes, sometimes also natural resins. The protection afforded by those is minimal, but it was all they had - and to the naked eye, it does look like a great protection, since oiled wood has an entirely different appearance, and water droplets pearl on it instead of visibly wetting it.

This is a concept that is known in science as "face validity" - you look at an approach and your intuition tells you that it must be working, so you are likely to conclude that it works. Scientists try to prove that methods work independently of their face validity, but cooks and woodworkers rarely do so :) And since there is a tradition, which convincingly does something, people continue applying it and promoting it.

The oil protects the wood from drying out not from physical harm. Drying out can lead to the board warping, cracking, or even falling apart if you have a cutting board that is made of multiple pieces of wood.

I have solid single piece wood cutting boards that are decades old and still going strong with no oil but I've also had cutting boards split in half after only a few uses when someone ran it through the dishwasher, which removes the oil (detergent) and dries it out (heat drying cycle).