The most pressing problem in the story - certainly for the main character, Digory - is how to heal his mother, who is terminally ill. We are not told exactly what it is that she suffers from, but it is likely something like cancer. Digory loves her dearly but feels helpless about the situation. It seems there is no cure for her in this world, but, inadvertently, by means of magic rings created by his uncle Andrew, Digory is catapulted into other worlds, finding at last a magical apple with the power to heal. He is helped in this by Aslan, the great creator of the beneficent world of Narnia. He brings the apple back with him to this world, and when his mother eats it she is cured.Therefore an excursion into magical other worlds provides the solution to the central problem of the book.
Another problem develops in the course of the story, when Digory and his friend Polly first encounter a witch called Jadis in another world and end up bringing her back with them when they return. She is evil and powerful and threatens to wreak all manner of destruction, just as she did in her own world, Charn. Digory and Polly don't know how to get rid of her. They try to get her back to her own world with the magic rings, but she ends up accompanying them to Narnia, and threatens that newly-created world also. Although he did so by accident, Digory has to atone for bringing her to Narnia. Therefore he is sent by Aslan on a quest to find the magic apple to plant in Narnia, which grows into a beautiful tree that will protect Narnia from the Witch for a hundred years or so (although she re-appears in the next Narnia adventure, The Lion, the Witch, and the Wardrobe).Therefore, the magic apple helps solve two major problems in the story.
A third problem is posed by Uncle Andrew. Although not as evil or powerful as the Witch, he is a mean-spirited, unfeeling, selfish man who thinks nothing of trying out potentially dangerous experiments on others, even children, like Digory and Polly when he sends them into other worlds. However, he is punished in the course of the story; at first he is terrorised by the Witch, then later has some unpleasant experiences when tranported to Narnia. He becomes chastened and humbled by the end:
Uncle Andrew never tried any Magic again as long as he lived. He has learnt his lesson, and in his old age he became a nicer and less selfish old man than he had ever been before (chapter 15)
Uncle Andrew, then, changes wholly for the better. He is not magically transformed; he has to learn the hard way, but he does learn. This solves the third major problem in the story.