This correlation is mentioned in two different sections. It's first mentioned on page 547 and later mentioned on page 704.
The section on page 547 explains that when countries saw attendance fall at football games—or soccer, in America —they also saw the rise in the hooligan culture of fans. This, in turn, drove more people away from the sport and created a vicious cycle; when violence spiked, people generally stopped coming to the games, leaving only the fans who were considered hooligans in the audience. With this spike of violence came a rise in skinhead, side, and "fanaty" (fanatic) culture. Gangs began recruiting among the people who were left attending these games, which deterred attendance further. The author says that "the most vertiginous fall was reserved for the zone of the greatest social turbulence," which means that when a country's social climate was in upheaval, there were larger drop-offs in attendance.
On page 704, the author talks about factors that help drive this correlation. For example, when men are unable to find jobs due to deindustrialization, a country is likely to have more incidents of sports fan's violence. The author uses many different incidents from a variety of countries to show how these metrics are linked. Though correlation doesn't imply causation, it does seem possible that he's onto something and that some insight into social and political unrest can be found by looking at sports culture.