To get a good idea of the general tone of The Jungle Book, look no further than the first line:
It was seven o'clock of a very warm evening in the Seeonee hills when Father Wolf woke up from his day's rest, scratched himself, yawned, and spread out his paws one after the other to get rid of the sleepy feeling in their tips. (1.1)
It's all very casual, a wolf waking up and stretching, and the voice sounds like grandpa telling you a story around a campfire.
Our unnamed narrator always wants us to be comfortable, even when discussing difficult matters. For instance, when Mowgli must live in the jungle alone, we're told "he was not always alone, because years afterwards he became a man and married. But that is a story for grown-ups" (5.113-5.114). Again, it sounds like a grandfather speaking to a child. Hey, kid, I'll tell ya when you're older, but for now, don't worry your little head.