There aren't too many kids who don't like animals, which makes stories about them a great way to communicate truths about human nature to young readers. All kinds of animal stories have been written, and a lot of them are just about animals doing animal-y things, but some of the best aren't really about animals at all.
The Wind in the Willows, for instance—sure, Ratty and Mole have whiskers, and Badger is as grumpy as any badger whose hole does not include a well-stocked library, but they're essentially human characters with human emotions and human ambitions who grow the way humans ought. Even Mr. Toad is no more animal than Oliver Twist or Laura Ingalls.
Okay, you might say, but that's not really an animal story, then. What about Black Beauty? or Bambi? They're written from the perspective of real animals who don't talk (at least, not to people) or wear clothes or eat buttered toast; how can they be about human nature?
In the case of Black Beauty, the horse narrator has a surprisingly firm grasp of human emotion, and understands the behavior of her owners to an extent many homo sapiens would not. As for Bambi, while he acts like a real deer, he's also sentient and able to have relationships with his fellow forest animals that closely resemble those enjoyed by members of the human race.
Another kind of animal story is really about people, and simply uses animals as the hook, or as a foil for the humans in the story. The Black Stallion or Billy and Blaze books, for instance, are more about the boy heroes than the horses, though the bond between animal and rider is also explored. Jim Kjelgaard's dog stories are similar, chronicling the adventures of boys with their beloved dogs.
As with any genre, animal stories should be read primarily to be enjoyed. It is only then that the deep wisdom found in the best of them can speak to us as readers, can open up our minds and hearts to see more than just furry or scaly creatures do things they may or may not do in the real world. Besides, who's to say there isn't a talking monkey or cat somewhere in the real world? As far as books are part of the real world, there are plenty such animals, and plenty of opportunities to meet and be changed by them.