Understanding history is important to assist in appreciating the past. Presently there is ample of data storage available at prices cheaper than before. Not only is it important to preserve ancient dialects to appreciate a societies' history, but the methods for doing so are cheaper and more readily available today.
Ancient dialects can usually tell you a lot about the history of a place. It is important that we preserve heritage such as dying dialects so that we and future generations can continue to learn from them. It's a shame to let good history go to waste, it shouldn't be allowed to die.
Ancient dialects that were spoken hundreds, even thousands of years ago are either dead or dying rapidly due to the simple fact they are no longer in use. many Native American tribes are working to preserve their languages for future generations, as are many other cultures around the world. They do not see these 'dead' languages as just relics of a forgotten past but more essential parts of their identities and their cultures.
Cultural transmission works (statistically speaking) across genetically based boundaries only in case of languages with high prestige and/or official status in a multi-ethnic territory, like Greek in the eastern and Latin in the western part of the Roman Empire or English in the former British colonies. But if you find a relatively small community, occupying a well-defined territory and speaking a language spoken by nobody else in the world, you can reasonably assume
that they share a common "tribal" origin. A native speaker of Spanish can be of any descent: Germanic (as the Iberian Peninsula was once inhabited by Visigoths), Celtic, Arab, Berber, American Indian, African, Asian (Cuba had a large wave of immigrants from China), etc., but if in Mexico you visit a Nahuatl-speaking village, you can be almost sure that all or almost all of its inhabitants descend from the Aztecs, or at least from those Indian tribes that found themselves withing the confines of the Aztec Empire in 15. century CE.