Language is the natural result of two human brains attempting to communicate.
That's pretty much it.
We're looking at the wrong question. It's not how, but why. If all the food in the world was no more than 6 feet off the ground, would we have giraffes?
Human language originated as humans originated. While the line that led to giraffes was getting taller and the line that led to apes was getting bigger, we were getting smarter. For some reason, it was to our advantage to become the second or third smallest primate, become relatively hairless, and compared to our simian cousins, not that physically impressive. While the rest of the animal world was growing physically specialized, we became mentally specialized.
It's a rough world out there. Leopards are a bitch. They just run up and grab the smallest person in the bunch and drag him away in the dark. No future at all in that. Human survival, even to this day, has always depended upon the cooperation within a group. Other animals have family groups and some of them have cooperative social structure, but none are as complicated as human society. In order to cooperate, we have to communicate. Whether it's blinking our eyes, making shapes with our fingers, or talking, we can make a language our of any controlled motion.
Animal calls are sufficient for animal needs. Once you've covered "get away from me," "where are you," and "watch out for the leopard," there's not a lot animals have to say to each other. What else do they need to tell one another in order to survive another day. It's not that easy for humans. We can't outrun the leopard and we don't really want to leave one of the children behind while we get away.
It is critical to human survival to be able to communicate quickly and clearly about everything in the environment. It is the key to cooperation. The part of the brain that controls the fine motion in the fingers, also controls the fine motion in the tongue. I don't think this is a coincidence. Hand signs accompanied by sounds can be an excellent communication system. It doesn't have to be an all at once development. We've already got use for fingers, tongues and lips. Re-purposing them for something of immediate benefit is an easy move.
One thing that the humans have never faced, at least as a species, is loss of habitat. With our big human brain, and our cooperative groups, any habitat in the world is ours. We live in the desert and above the Arctic Circle. Let a leopard try that.
It's not really a mystery of how we developed language. It developed as we did. It was critical to our survival and without it, there would be no one to talk about it.