Exposition is just a fancy way of saying background information, and every story has background information the reader needs to know to proceed. It's the "who," "what," "where," "when," and "why" of the story. Every now and then, we even get a "how" thrown into the mix.
In "Rikki-tikki-tavi," the exposition is contained in the beginning of the story. We learn who Rikki-tikki is and how he comes to be with the British family. We also discover a lot of useful information on the nature of mongooses—information to help us understand why Rikki-tikki fights against the cobras. We also meet the other players of the story such as Darzee, the British family, and Nag and Nagaina.
Roughly speaking, the exposition ends once we have all this information. For this story, that's when Nag and Nagaina slither away after having confronted Rikki-tikki. At this point, we know who almost every character is and what their place is in the story. We also understand what the conflict will be, so now it's time to get the exciting stuff underway.
The rising action is the part of the story where the action begins to, you know, rise. Pretty accurate label, no? Another way to put it: the conflict heats up and the difficulty increases for the protagonist.
In our short story, the rising action consists of Rikki-tikki's battle with Karait. Since it's the mongoose's first kill, it helps his build his skills toward the inevitable scuffles with the cobras.
Rikki-tikki's vendetta against all snake-kind is complicated by the fact that the cobras have wants and desires of their own—like the desire to raise a family. Like many expectant parents, they're no longer content with their little garden hole; they want a second bedroom, and maybe even a playroom or family room.
To do that, they decide to kill the entire British family, meaning that Rikki-tikki has to step his game up—stat.
The climax is the turning point in the conflict. Everything about the story changes, and there's no going back to the previous status quo set up in the exposition.
In "Rikki-tikki-tavi," that's the battle with Nag. The battle is Rikki-tikki's most daunting challenge, one that almost kills him. Also, the battle changes the relationship between our protagonist and the antagonists. Before this, Rikki-tikki was a nuisance to the cobras, but now he's a genuine threat to Nagaina. The same change happens in Rikki-tikki. After he kills Nag, his confidence grows. He realizes he has to deal with Nagaina now, as she'll be out for blood.
The name "falling action" is a bit misleading. It might sound like the action is dying down at this point, but actually the action can be just as intense as ever. What's falling is the conflict between the antagonist and the protagonists, and it falls toward the resolution.
And in this story, the falling action constitutes the battle between Rikki-tikki and Nagaina. First, Rikki-tikki has the upper hand as he destroys Nagaina's nest. But then Nagaina regains it when she attacks the family on the veranda. And when Rikki-tikki comes back with the egg, things switch again. Of course, Nagaina gets the egg back, so that's another switcheroo. See? This action isn't tapering off at all. Rather, the conclusion inches closer while the tension ratchets up from all the reverses in the power struggle.
The falling action ends with that final bit-o-suspense before we hit resolution: Rikki-tikki and Nagaina both enter the cobra's den. Who will emerge?
Finally, we reach the resolution. The conflict ends, the plot threads are tied off, and we discover the fate of those involved.
"Rikki-Tikki-Tavi" plays things safe and gives us the happy resolution. Rikki-tikki emerges from the den, proclaiming Nagaina dead. The animals rejoice and peace comes to the garden. Rikki-tikki stays with the family and lives his days out as their protector.
And the cobras? They never enter the garden again.