What is language, then, if it can describe the way we process actions as well as the way we manipulate words? Understand from this perspective, language is not a method of communication, per se, but a rather method of computation. Other animals clearly communicate with one another, sometimes in fairly elaborate ways. Whale sing, monkeys howl, birds chirp. Lizards bob their heads up and down to communicate, and some squid do it by regulating the colouration of their skin cells. But none of these processes can be explained by language.
What makes human language unique is not that it allows us to communicate with each other, but that it allows us to do so with infinite variety. A monkey can scream to warn its troopmates of an approaching predator, or alert them to a cache of tasty food, but it can't communicate something like "doesn't that hawk have a funny looking beak?" or "with a little salt, this fig would taste divine". It certainly can't create nonsensical yet understandable sentences like “Colourless green ideas sleep furiously”.
No, only humans can utter that sort of grammatical nonsense.