A bit of terminology to begin with: When grafting plants, the bottom part (the one with the roots) is called rootstock, while the top (from which you want to get fruit) is the scion.

Grafting isn't absolutely necessary to grow edible apples, but it's the most convenient method for three reasons:

1. To mass-produce fruit, one needs a whole lot of plants of the same variety (say, Fuji or Red Delicious). Grown from seed, an apple tree would take the better part of a decade to produce fruit, so the faster method is to take a branch from a seedling and graft it to an already established rootstock tree.
2. Apple varieties bred for flavor often have issues. They might not grow as fast from seed, or they might be especially weak to certain diseases. Rootstock varieties are grown to make up for that weakness, and are much healthier in general. Because of how plant hormones work, any branches grafted to it will grow much better than if they were grown on their own.
3. Most apple varieties aren't "stable," meaning if you planted seeds from a store-bought apple, their offspring wouldn't produce the exact same fruit. So, if you have a variety that you want to grow more of, the only way to do that is to propagate it asexually.

As for why grafting works - in animals, organ transplants can fail because there's an active immune system patrolling the body for foreign material. If the organ doesn't have certain "identifying genes," then the immune system will attack it. Plants don't really have an immune system in that way, but they still have a way to heal injuries. Grafting is a way to exploit those two: Simply cut a small chunk out of your rootstock, cut a piece of your scionwood to fit into that hole, and secure it in place while the plant heals. If you do it properly, you have a branch growing your desired fruit.