I do have form on this topic - disappearing languages - but it's a great help to find a comment which states the case and marshals the arguments efficiently and without the rancour that marks my outbursts. So I quote -

Aside from literature (most languages do not even have a written form)
the loss of even a single language entails the loss of a wealth of songs rituals and histories -- an entire culture, so to speak. Effects on cultural identity, education and regional and national politics are profound.

Then there's the dissolution of indigenous cultures' store of knowledge about local plant and animal life -- which, especially in the Amazon area and Black Africa are relatively unmined by biology and medical science.

And of course there's the loss to linguistics. Typological studies reveal the vast diversity that exists in the phenomenon of human language, and the often alien-seeming modes of expression and intricacies of grammar and phonetics. A minority of the world's languages has been studied in enough detail to enrich our understanding of how the mind works in organising thoughts and mental constructs into language. Australian Aboriginal languages, Siberian languages and some Native American languages are often cited to illustrate the often fundamental differences with more familiar languages (European languages, Chinese, Japanese), but there are bound to be more fantastic languages out there.

Losing languages is also a severe impediment to tracing the spread of human populations and large-scale migrations, since accounts of these are based on comparing evidence from archeology, history, genetics and linguistics. Charting the spread of languages is key in understanding historical migration patterns; the settlement of the Americas or the Austronesian homeland question are fine examples. While the interrelations between Indo-European languages are generally well understood, the patterns are generally less clear for any other language family out there. Each undocumented language that dies out before it has a chance to be studied means that the number of clues about linguistic kinship is greatly diminished (imagine reconstructing Indo-European without Sanskrit, or Greek).

There are a number of organisations that attempt to document languages and revive them. The most prolific one that I am aware of is Wycliffe, which study small and endangered languages in order to translate the bible into them. I'm not so sure that Wycliffe are doing an altogether laudable job -- from a linguistic point of view, that is. Nonetheless, even with their limited linguistic focus, they're doing invaluable work in preserving at least some of the diversity of human languages that we don't know about yet.