As I explained in a previous post, we have no idea, really. We can at best establish some very tentative point after which language might have shown up, and we can establish more confidently a point before which language must have existed (roughly 10,000-15,000 years before present, if our estimate of the time for Proto-Afro-Asiatic is in the right ballpark), but we really can't narrow the time frame down more than that.

The how doesn't have a satisfying answer either. Famously, in the 19th century the Paris Linguistics Society declared the origin of limits for its meetings, and several linguists continue to follow their lead in that. Some scholars who are, depending on your perspective, braver or foolhardier than their peers, have proposed several scenarios, but the fact remains that there's just no way to confirm any of them.