First:

He begins the story grieving, without hope that his’ mother will soon die (and he has good reason to expect her to die as all the evidence supports that her condition is hopeless).

… and if the reason was that they were looking after your Mother—and if your Mother was ill and was going to—going to—die.

By the end of the story, his mother has undergone a miracle and is no longer going to die. Even before the miracle has taken place, Digory has gone from a place of hopelessness to a place of hope:

For the rest of that day, whenever he looked at the things about him, and saw how ordinary and unmagical they were, he hardly dared to hope; but when he remembered the face of Aslan he did hope.

The dramatic change has come from Digory’s encounter with Aslan; Digory realizes that Aslan is able to do what Digory could never do.

Second:

Digory begins the story needing to control his situation; he ends as someone who can relinquish control of the situation.

From my reading, the book can essentially be divided into two halves with a roughly chiastic structure making up each half.

The first half could be seen structured in the following manner:

i       Digory & Polly in tunnel on rafters.
        ii      Uncle Andrew’s forbidden study
                iii     Wood between the worlds
                        iv     Charn
                                v      Hall of images
                        iv2   Charn (with Jadis)
                iii2   Wood between the worlds (with Jadis)
        ii2    Uncle Andrew’s forbidden study (with Jadis)
i2     Polly returns home through tunnel on rafters

The second half could be seen structured in the following manner:

i       London streets (fight at the lamppost with Jadis)
        ii      Wood between the worlds (with Jadis)
               iii     Narnia & Meet Aslan (with Jadis) - raise issue of dying mother
                      iv     Flight on Fledge from Narnia to Garden
                             v    Encounter in Garden (Jadis' temptation)
                      iv2    Flight on Fledge from Garden to Narnia
               iii2   Narnia & meet Aslan again - given cure for dying mother
        ii2    Wood between the worlds (with Aslan)
i2     London streets (post-fight without Jadis)

 
The final part of the second chiasm is unnecessary - Aslan can transport the children and Uncle Andrew without the rings, so there is no reason to return to the Wood Between the Worlds and they could have re-appeared on earth in Digory's house rather than back on the street where the previous incident had occurred. Lewis, nevertheless has the ending echo the earlier part of the story to ensure the chiastic structure.

At the center of each of these chiasms is a temptation scene.

The first time, Digory is tempted to strike the bell with the hammer.  He believes that if he does not strike the bell that he is fated to go mad.  Even though he knows that striking the bell will be dangerous, he convinces himself that he is enchanted and has to go through with it because he does not want to be driven ‘batty’.

The second time, Digory is tempted to eat the apple himself or to take a stollen apple to his mother.  Digory resists the temptation this time.  He knows that by not giving the stollen apple to his mother, he might be subjecting her to an inevitable death, but he does what is right rather than seeking to control his situation.

Note also, that in the first half, Jadis only appears in the story after Digory gives into the temptation and strikes the bell at the focal-point of the chiasm. In the second half, Jadis disappears from the story (running away to the North) when Digory stands firm and resists the temptation.

On the flight back to Narnia, we are given some of the reason for the change:

He was very sad and he wasn’t even sure all the time that he had done the right thing; but whenever he remembered the shining tears in Aslan’s eyes, he became sure.

It is because he has encountered Aslan, that he knows he does not have to be in control of everything.  This goes back to his earlier encounter with Aslan.

When Aslan and Digory first meet, Digory wants to secure a cure for his mother:

“Please—Mr. Lion—Aslan—Sir,” said Digory, “could you—may I—please, will you give me some magic fruit of this country to make Mother well?”

and he is also defensive:

Digory slipped off the horse and found himself face to face with Aslan. And Aslan was bigger and more beautiful and more brightly golden and more terrible than he had thought. He dared not look into the great eyes.
…
He had had for a second some wild idea of saying “I’ll try to help you if you’ll promise to help my Mother,” but he realized in time that the Lion was not at all the sort of person one could try to make bargains with.

Once Digory lets go of his defenses and allows himself to really be transparent with Aslan, his world is unmade:

But when he had said “Yes,” he thought of his Mother, and he thought of the great hopes he had had, and how they were all dying away, and a lump came in his throat and tears in his eyes, and he blurted out:

“But please, please— won’t you — can’t you give me something that will cure Mother?” Up till then he had been looking at the Lion’s great feet and the huge claws on them; now, in his despair, he looked up at its face. What he saw surprised him as much as anything in his whole life. For the tawny face was bent down near his own and (wonder of wonders) great shining tears stood in the Lion’s eyes. They were such big, bright tears compared with Digory’s own that for a moment he felt as if the Lion must really be sorrier about his Mother than he was himself.

This is the turning point where Digory is changed.  He can relinquish control and obey because he learns Aslan’s character and knows that he is trustworthy.