One of my fond memories of college involves my English teacher and how she went on a tirade about how the first humans invented the wave (as in waving to your friend to say hello). How did we come up with that gesture?
Now I'll extend that sense of wonder to games and sports.
First, there's sports like running and boxing. Now I can imagine this would come up naturally. Two guys wonder which of them is the fastest and viola, the first race. Or perhaps they were pissed off at each other, started fighting, and the rest of the people in the vicinity started watching.
Then there are sports where it gets a bit more complicated. Say basketball or soccer. Now in both sports, the concept is the same: sink the ball into the goal. Of course it gets stranger when there's an arbitrary limitation, such as not touching the ball with your feet in basketball, or not touching the ball with your hands in soccer (and then there's the concept of the goalie in soccer).
I think it gets weirder in sports like American football or baseball. Now these are sports which are definitely human constructs. They both have highly specialized rules and roles. Same goes with something like chess. For me it's more likely to come up with the concept of the game Go than it is with the game chess (although the former is more difficult to master).