The answer is that we really have no idea, and we can't ever know. Some people will point to the development of anatomically modern vocal tracts (or at least, the bone structure that can support them) and give that as the date. Perhaps they're right, but language doesn't necessarily use the auditory channel, and it's perfectly possible that the first language was a sign language.

Linguistic reconstruction can't hope to help us--anthropologists have established that anatomically modern humans existed 100,000 years ago (if not earlier). Proto-Indo-European is one of the oldest reconstructed languages we have evidence for (AFAIK), and the date for PIE is probably about 7,000-8,000 years ago. Indo-European is lucky to have some very early attestations in the form of Anatolian, Greek, and Sanskrit and to be generally very diverse. While it's possible to do linguistic reconstruction with reconstructions as your data, it quickly becomes fairly difficult. As a rule of thumb, linguists sometimes say that the method can't get you further than about 10,000 years ago.

Basically, if gives you a date for the origin of human language, don't trust them, because there's absolutely no way to know.