How do you describe how someone showed bravery or courage in "Rikki-tikki-tavi" using the words demonstrate, exhibit, display, or illustrate?

Rikki showed courage in this story by facing down two cobras (and little Karait) in order to protect his family. All of the words that you have provided are synonyms. This means that they are similar enough that they could probably be used interchangeably.

Let’s take the first word, “demonstrate.” The first time Rikki showed bravery was by the way he reacted to being washed away and finding himself in a new place with the British family. They rescued him and invited him into their home, and he reacted with curiosity instead of fear.

He spent all that day roaming over the house. … At nightfall he ran into Teddy's nursery to watch how kerosene-lamps were lighted, and when Teddy went to bed Rikki-tikki climbed up too; but he was a restless companion, because he had to get up and attend to every noise all through the night, and find out what made it.

Using the word “demonstrate” you could write a sentence like this.

Rikki-tikki demonstrated courage by moving in with the family and exploring their house.

The next word is “exhibit.” It means “to show.” Rikki showed bravery by how he reacted to Nag and Nagaina, the cobras.

He was afraid for the minute; but it is impossible for a mongoose to stay frightened for any length of time, and though Rikki-tikki had never met a live cobra before, his mother had fed him on dead ones, and he knew that all a grown mongoose's business in life was to fight and eat snakes.

You could write a sentence about this incident using the word “exhibit.”

Rikki-tikki exhibited courage by standing up to Nag and not allowing Nagaina to trick him.

Since the word “display” is similar to “exhibit,” you could describe Rikki’s fight with Nag.

Rikki-tikki displayed courage when he searched for Nag and killed him before he could target the family.

Finally, that leaves “illustrate.” This word also means “to show.”

Rikki-tikki illustrated courage when he used Nagaina’s eggs against her and followed her into her hole.

In the end, Rikki-tikki defeated all three snakes, and kept the garden free from snakes from thereon.