The story is that you're essentially going on an excursion into the wilds of the Bioluminescent Forest to explore its natural wonders. Given the structural framework of Animal Kingdom, that's about as far as any attraction story goes in that park. It's not a park about Disney Storytelling™ in the "traditional" sense, it's about creating the experience of going out into nature and building a connection with it through experience. This is similar to Kilimanjaro and even Flight of Passage - you're going on an excursion in an exotic, animal-rich environment to be awed by its beauty and moved by its splendor. The alternative to this storyline within AK is more like what we experience in Kali, Everest, and Dinosaur - we enter the natural world in ignorance of its order and through our experience we are humbled by its power.
You can totally debate the success of its execution - it's fair to say you think the Safari or FoP deliver better on this premise, and I tend to agree that the ride feels a bit like it's over before it's truly begun, though I do enjoy what's there - but the story isn't lacking by the standards of that park. Animal Kingdom is meant to mirror a real-world-type experience through its rides more than it is meant to tell a "story". That if Pandora was a real place and you went on an expedition into the forest, this is what it would be like. No one complains about the "lack of story" on Flight of Passage even though it's basically the exact same as Na'vi River Journey - go on a unique excursion through the otherworldly natural landscape of Pandora courtesy of Alpha Centauri Expeditions - but I think that simply comes down to it being generally regarded as a more satisfying attraction experience. It offers neither more nor better story, just a more exciting experience.