Professional sports (as well as college sports which were then, as they are now, professional in many ways) came about largely because of the urbanization of the American population during the time known as the Gilded Age. Sports such as football (mostly college) and baseball (professional) joined sports such as boxing to become big business.
This was possible in part because there were so many people in the big cities. This made for a large fan base that could attend games. News about the games could be spread via telegraph and newspapers and magazines. This also helped to create interest in the games. The web of railroads made it relatively easy for teams to travel to play one another. For these reasons, sports became a big business during the Gilded Age.