To put it simply, Shere Khan is angry that the wolves have adopted Mowgli, as he hates humans and feels that Mowgli is his to kill.

This rage all stems from an inferiority complex. Shere Khan was born with a crippled leg and therefore scorned by his mother. He seeks to be the lord of the jungle as a way for making up for that physical flaw. He particularly loathes human beings since, with their advanced intelligence and knowledge of how to make fire, they pose a significant threat to his power.

Shere Khan was trying to attack humans when Mowgli was separated from his parents and then found his way to the wolf pack. The wolves took Mowgli in before Shere Khan could hunt him down. Now that the human boy is protected by the wolves, Shere Khan cannot kill him without breaking the Law of the Jungle and then suffering the wrath of the wolf pack. As a result, he is angry and holds a grudge against Mowgli that never abates even as the boy ages. His hope is to one day overturn the wolf pack by promising the younger wolves greater boons once their current leader, Akela, is out of the way and then convince them to turn Mowgli over to him, so he might at last satisfy his bloodlust.