Running was not so much "invented" as it is an innate part of the human design. It is an instinctual motion, as humans grow and change, for them to also evolve in the ways that they move. For instance, we do not really "teach" a baby to crawl. They instinctively grow and carry their weight to a point where they understand that if they want to get some where quicker and more efficiently, they also need a more efficient way to get there. And then the same is true when learning to walk and then to run.

However, we know that running has existed as part of typical human life basically since the beginning. The earliest humans were cave men who hunted and gathered for their food. And one of their most commonly used means of escaping predators and dangerous situations was good old fashioned foot speed. But running has not just been a means for escaping danger. Running was considered a sport as far back as the Ancient Greeks. In fact, it is considered THE oldest Olympic Sport, as for several years it was actually the only event in the ancient Olympics. And even from there, the sport of running evolved to include more events at different distances for participants to participate in. (However, at this point, running and the competitive sport of running was pretty much exclusively done in the nude!)

Sources:
http://ancientolympics.arts.kuleuven.be/eng/TC002EN.html