"As languages disappear, cultures die. The world becomes inherently a less interesting place, but we also sacrifice raw knowledge and the intellectual achievements of millennia."
Ken Hale, Massachusetts Institute of Technology, quoted in Davis, W. 1999.
Language and your cultural identity are very tightly coupled. Without language you essentially lose your identity, which is why its crucial to preserve your language. It's one of your primary identifiers of you and your culture. Wenying Jiang stated it quite well in her paper, "The Relationship Between Culture and Language" when she says your culture is like an iceberg. Your language is the tip which sits above the water. The rest which sits under the water is the rest of your culture. Your customs, your religions, your societal norms.
Link to her paper which is worth a read: http://203.72.145.166/ELT/files/54-4-3.pdf
Also consider the following:
"With the loss of the languages, all kinds of wonderful things that the speakers did with their languages have also vanished, for example, some of the greatest works of oral literature ever produced–the multilingual performances with different characters speaking different languages that was found in the Pacific Northwest.
The highly elaborate dances that accompanied the oral tradition are frequently also gone.
Large amounts of local knowledge about fauna and flora, ecosystem management, local place names, spiritual values, and so on are all submerged, altered or gone because the original languages that expressed these concepts are gone or no longer well understood."
source: http://newswatch.nationalgeographic.com/2009/11/15/0005_nati...
"As each language dies, many sciences - linguistics, anthropology, prehistory and psychology - lose one more precious source of data, one more of the diverse and unique ways that the human mind can express itself through a language's structure, vocabulary and idiom."
Linguistics is obviously most affected. The variety of sounds and grammar used in languages in huge, and we wouldn't know about it without being able to study a lot of languages. I don't know if that has any economic value[3], but then neither does paleontology, but we consider that pretty important, important enough to preserve the available data.
Really, as nice as it would be to keep these languages spoken, preserving a complete grammar and audio recording of individual sounds as well as longer spoken passages would be enough for most purposes. But the problem is that this is a lot of work, and languages are going extinct faster than we can document them.
[1] https://en.wikipedia.org/wiki/Language_preservation
[2] http://www.ogmios.org/manifesto/index.htm
[3] Well OK, some people get paid to invent languages for sci-fi/fantasy and a lot of linguistic data helps with that, but... that's not much.
This is a problem in contemporary linguistics- the "big" languages are really heavily studied and theories are developed to describe them accurately, nevermind that they may completely fail for, say, Xhosa or Straits Salish. The bias has been decreasing over the last few decades, but getting a really good idea of the real fundamental underpinnings of human language depends on all those minority languages not disappearing, undocumented, before we're done.
Of course, there are other reasons which are important to people working on other fields- languages are strongly tied to culture, and when a language dies, so does a lot of other cultural knowledge and practice that may be useful to ethnologists, psychologists, philosophers, etc.
A good overview: https://en.wikipedia.org/wiki/Linguistic_relativity
In addition to the above pure artifact preservation aspect language (when paired with a lot of text, this is not always the case, e.g. Etruscan language) can make available vast data about a community. When Asyyrian was decoded and the great libraries of tablets were discovered (e.g. http://en.wikipedia.org/wiki/Library_of_Ashurbanipal), our knowledge of the culture of the Ancient Middle East increased exponentially.