I have my own theory that nobody seems to have addressed which is basically that it was an intentional decision created by one individual, likely a leader of a specific tribe who decided it would be convenient to have a sound he could use to refer to something. So, for instance, he wants a term for "rock", so he simply comes up with a noise for it and gets the attention of the other tribe members by making that noise while interacting with the rock in some way. Eventually through pattern recognition abilities the others catch on and with that we have the first language with a grand total of one word. This process repeats many times, likely over many generations with more and more words added over time.

I haven't heard of anyone explaining why this idea wouldn't work. I've heard it said that humans couldn't have sat down and decided to make a language because they would have already needed a language in the first place, but that only applies if the idea was executed by a group rather than a single specific individual.