Language evolved over a few million years.
Its not that hard to see a group of human ancestors with no speech ability using whistles or grunts to communicate while hunting or letting others know a predator is around. Birds, monkeys and I'm sure other animals do this in todays world.
Those groups of people that could communicate danger, food, or hunting signals the best survived and so did those would could understand them. Over time they evolved and became more adept at using vocal signals and could probably mimick sounds like bird calls etc. Each step along the way it became closer to current human langauge. But even nowadays the !kung! bushman and some tribes in africa still rely on language that consists of clicks and whistles.
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