(Part 2) Here's another example.

Here is something told without illuminating details:

"For his wish, Midas asked that whatever he touched would turn to gold. It was granted! Excitedly, Midas went about touching all sorts of things, turning them into gold.Soon Midas became hungry. He picked up a piece of food, but he couldn't eat it, for it had turned to gold in his hand! I'll starve,' moaned Midas."

With illuminating details: (Nathaniel Hawthorne, "The Golden Touch")

"He took one of the nice little trouts on his plate, and touched its tail with his finger. To his horror, it was immediately transmuted from an admirably fried brook-trout into a gold-fish, though not one of those gold-fishes which people often keep in glass globes, as ornaments for the parlor. No; but it was really a metallic fish, and looked as if it had been very cunningly made by the nicest goldsmith in the world. Its little bones were now golden wires; its fins and tail were thin plates of gold; and there were the marks of the fork in it, and all the delicate, frothy appearance of a nicely fried fish, exactly imitated in metal. A very pretty piece of work, as you may suppose; only King Midas, just at that moment, would much rather have had a real trout in his dish than this elaborate and valuable imitation of one. 'I don't quite see,' thought he to himself, 'how I am to get any breakfast.'"

By describing the details of the fish so carefully, the author makes us "see" both how wonderful and how horrifying the Golden Touch is. Figuratively, he illuminates--shines a light on--what it would really mean for someone to have the Golden Touch.

The story, told with just the bare facts, is a good story. With the "illuminating details," it becomes a much better story.

(Part 1) "Illuminating" literally means "shining a light on." In the context of a story, an illuminating detail is something that makes the listener say "Oh, now I see." That is, a specific and small detail that really helps you understand the story. It shines a light on the picture in your head.

Here is a something told without illuminating details:

"It was very cold, far below freezing. It was -60°C. But the man did not know it."

Here is something told with illuminating details (Jack London, "To Build a Fire.")

"As he turned to go on, he spat speculatively. There was a sharp, explosive crackle that startled him. He spat again. And again, in the air, before it could fall to the snow, the spittle crackled. He knew that at fifty below spittle crackled on the snow, but this spittle had crackled in the air. Undoubtedly it was colder than fifty below -- how much colder he did not know."

The spit crackling is a small detail about how cold it was, but since I've never seen it happen, it makes me realize that this is colder cold than I've ever experienced.

Well, you need something in front of story, like „a“ or „the“. Illuminating is just saying that the story is told in a way that opens up, or better sheds light on certain details. Don`t forget, to illuminate just means to lighten something up (like with a lamp, flashlight, that sort of thing). 

Plus, I think your „in“ is misplaced because it sounds off to me. No offense, but the sentence could be improved a bit. „She tells the story, illuminating the details“ or „with illuminating detail“ sounds a little better to my ear (don`t ask me why).