The short answer is because it requires very little equipment and because the British Empire had some reach to it. The first rules as we'd really recognize them today (the Cambridge Rules) were written up in the 1860s when the sport was still played exclusively in England by amateur teams. It was a sport mostly accessible to the working class due to needing nothing more than a ball and some open space (as opposed to cricket, which also needed bats and protective gear and a wicket). A few things happened in a relatively short span:

1. The first international game between England and Scotland in 1872.
2. The growth—and eventual acceptance—of teams hiring "football professors" (what we're call pro players) in the 1880s.
3. The spread of English people across the globe for non-football reasons who ended up bringing the game with them to continental Europe and South America.

These things allowed the sport to spread rapidly among both amateur and professional ranks, and the rise of national play allowed other countries to try to take a shot at beating England at their own game in a time when British imperialism was a strong force in global politics.

I heavily recommend reading a book called Inverting the Pyramid by Jonathan Wilson. It claims to be a history of soccer tactics, but the first few chapters also cover the proliferation of the sport to continental Europe in rather a lot of detail (including how some countries and cities deliberately hired English managers to teach them to play specifically so they could try to beat England and English teams).

I'd also recommend the Netflix series The English Game. It's a drama that captures the essence of how high society and the working class came together over football, how the FA reacted to early professional players, and how football teams could become a point of pride for a town. It's one of those things where the facts aren't necessarily right, but the overall vibe and general thesis is on point.

ETA: I say "accessible to the working class" in the first paragraph, but I should be clear that I mean that only in the base financial sense. Football as it started was—like many sports at the time—played by the upper class who had more time (and energy, since they weren't manual laborers) for leisure, and in the early days it really was protected as the gentlemen's sport. Again, see Inverting the Pyramid for a great discussion of how early football tactics rerflected the social heirarchy of English boarding schools.