This is what you call an etiological story. An etiological story seeks to explain why certain things are the way they are. In this story, Kipling is explaining a few things. First, why animals live in the jungle. Second and more importantly, why the Ethiopian has dark skin and the leopard has spots.
According to the story, the Ethiopian and the leopard were friends in a mythological place called High Veldt. They hunted the other animals, but the animals decided to move into the jungle. When the Ethiopians and leopard were hungry they left at the advice of the wise baboon, named Baviaan. When they arrived at the jungle, everything was different. The animals, such as the zebra, changed. What were the Ethiopians and leopard to do? They heeded the words of the baboon, and they changed as well. The Ethiopian darkened his skin and the leopard gained spots, to fit in and to hunt.
This is how the story ends:
Oh, now and then you will hear grown-ups say, "Can the Ethiopian change his skin or the Leopard his spots?" I don't think even grown-ups would keep on saying such a silly thing if the Leopard and the Ethiopian hadn't done it once do you? But they will never do it again, Best Beloved. They are quite contented as they are.
As you can see the story seeks to explain the current state of affairs.
We can actually look at this story in another way, based on the last few sentences. Can change take place? The answer is clearly yes! In fact, the quote, "Can the Ethiopian change his skin or the Leopard his spots?", is taken from the Bible - Jeremiah 13:23. In Jeremiah the point is that people cannot change, but in this short story, the point is that people can change. This would be a great discussion.