By Using Language Rooted in Andes, Internet Show’s Hosts Hope to Save It

Preserving dying languages is not unlike basic archaeology, preserving our past so that future generations can understand it, in a way that isn't superficial.

The way you put it, "the emotion of the people who feel it's important", is more sufficient justification, unless you propose it's better to sacrifice any human experience or paradigm that doesn't fill someone's coffers.

Studying another language, especially a radically disparate one, reveals nuances in your native tongue that are hard to come upon otherwise. If you accept even a weak version of the Sapir-Whorf hypothesis, then it's desirable to be aware of these nuances, as they may color your understanding of the world.

As far as the sciences are concerned, seeing alternate ways to encode information into language is extremely useful for people working in knowledge representation, ontology building, etc.

EDIT: I would also say that ""it gives us different ways of thinking" is all the justification needed for language preservation. That sounds pretty valuable to me.