In the US, I believe it was a confluence of at least three major trends. The first, urbanization, involved the massive growth of large cities that coincided with increased immigration as US capitalism was beginning to hit on all cylinders. Concurrently, a panic regarding masculinity and class engulfed the country. While robust and sturdy working class immigrants were flooding the cities, the traditional white owners of property and businesses, as well as the professional class that was emerging (lawyers, doctors, professors, clerics, bookkeepers, etc), were seen as less virile (as an analog, middle and upper class white women saw a revolution in how they dealt with childbirth, because doctors and society believed they, unlike the immigrant working class, could no longer tolerate the pain of birth). Third, mid century changes in collegiate social life had seen an explosion of sporting activity among college students. Being with gymnastics and rowing the 1830s (if memory serves me right), college students began playing organized sports. This was something of a boon to college presidents as it provided a somewhat organized way for students to interact with each other (though it should be noted that college authorities were, at the beginning, almost always wary of anything college students did). By the 1870s, baseball and primitive forms of football had been developed (and became useful in attracting the donations of alumni to college and universities that were, at the time, perpetually in debt). at this time, one can read the speeches of almost any college prudent and come away thinking that organized support and physical education were amongst the highest callings any person could have. It was to be the renewal of the old stock of white middle and upper class families while, at the same time, showing the immigrant working class what 'American' vigor and virility looked like. Organized professional athletics were a means to perpetuate both the careers of former college athletes and, at the same time, show case the athletic prowess of the white race.

And they're pretty fun to play and watch - especially while drinking and gambling (incidentally, the relatively high wages for some workers in the US at the time might have helped on the later front. I know that newspapers across the country carried college football being lines t 1890s and wouldn't be at all surprised if baseball and other sports has a similar trajectory).

Sorry for the long and rambling answer - such are the travails of whiskey and history!