There were a series of events that allowed football to develop into the popular sport it is today. To understand how this worked you need to understand what early 20th century football was. The game at this point was essentially rugby. The forward pass was illegal. Teams always lined up in what we would now consider a "goal line" formation. There were only 3 offensive plays: run right, run left, and run straight up the middle. On each of these plays the entire offense would create a wall and push forward, the entire defense create another wall and push back. A "win" on a play for the offense was a 2-3 yard gain, a win for the defense no gain or a small loss. The game was played mainly by football clubs in city neighborhoods and at universities by students in their spare time, mostly in the Northeast.

Because of the formations used, basically getting everyone into a small cramped space and pushing each other it was easy for someone to slip and fall and be trampled, particularly on kick returns where a formation called "the flying wedge", which looks like it sounds, was used. With fatalities mounting Teddy Roosevelt called on several University Presidents to reform the sport. The process took a while, but the main reforms were creating a neutral zone between offense and defense, making the forward pass legal, and banning mass formations like the wedge. This opened things up a bit and helped tone down the violence. The advent of the first major football star also helped. Jim thorpe was extremely fast, and instead of running headlong and getting what he could get he would run one direction, get the defense to follow him and reverse field and outrun the defense in the other direction to find open ground to run forward with. Pretty standard stuff today but at the time it was fairly revolutionary. As an aside he nearly ended Dwight D. Eisenhowers military career doing that as Eisenhower blew out his knee trying to tackle him and the Army nearly refused him a commission after graduating West Point. Thorpe was an Olympian in 1912 and a major figure in early pro football.

The game was already fairly popular, with college games drawing what pro baseball games did, partly because of the college affiliation and partly because of the scarcity, with teams playing much shorter schedules than pro baseball teams. Several pro leagues were tried, and the current NFL formed in 1920. The current dynamic of pro teams in larger cities, and college teams in smaller towns using the college affiliation to draw crowds was established. Red Grange played on one of the clubs that formed the NFL, and his name draw helped give it credibility. Jim Thorpe, the next major football star went from Illinois to the NFL after defeating Michigan in major game with the collegiate championship on the line. The combination of college pride drawing people in to that game and building stars that went on to the NFL was a potent one. Baseball didn't have anything like this, baseball players were no names when they went pro, football had two leagues that did not compete and were able to build off each other.

The next major shift for pro football started in the 50's and 60's. The forward pass went from being a legal and occasionally used play to a staple in that period. Early forward passes were all what we now think of as "hail Mary" plays. They were very long attempts that usually failed, but sometimes succeeded with dramatic results. In 1946 one of the greatest QB's of all time, Otto Graham was paired with one of the greatest coaches of all time, Paul Brown and they began the modern passing revolution. From 1946 to 1955 their teams played in 10 straight championship games, winning 7 of them. Because they attempted more passes, and completed more of them they opened the game, made it more exciting by increasing the number of dramatic "big plays", football's equivalent to the home run, and made the running game more interested because with the players spread out a little more on the field there was more running room.

The year Graham retired Johnny Unitas was drafted, and he became a star passer. In 1958 his Baltimore Colts played the New York Giants in the NFL Championship game and NBC decided to televise the game nationally. This was a fateful decision because the game was almost stacked to be an excellent contest. Each team had 6 future hall of fame players lined up, including Unitas who many feel is the greatest QB ever. Additionally, the Giants, who played in the largest media market in the country, had 2 future Hall of Fame coaches as assistants, Vince Lombardi and Tom Landry.

The star players in the game give a good idea of one of the NFL's most potent draws: the stars are already regional stars before they ever make the league. Frank Gifford of the Giants was from California and attended USC, so Californians had a player in the game. Giants defensive captain Sam Huff was a West Virginian, giving Appalachia representation. Don Maynard and Raymond Berry were Texans, and Johnny Unitas was from Pennsylvania. Rosey Brown of Virginia, Emlen Tunnell and Lenny Moore of Pennsylvania, and Jim Parker of Georgia were all black players in the integrated league. This gave fans from a lot of different areas someone to cheer for even if they weren't from NY or Baltimore.

The game was very close, with a dramatic finish and a controversial measurement for a first down late in the game that was the subject of much discussion for years after. It featured the NFL's best players and teams highlighting football played at the highest level in front of a national audience for the first time. It was a huge hit and the NFL has done nothing but grow and expand since.

Unitas and Gifford were stars in the league for years after the game. Landry and Lombardi went on to become championship head coaches and household names. Lombardi in particular is important in league history because of the way his Packers teams dominated the league in the 1960's. If Graham and Unitas created the current pass first football concept Lombardi sealed it. His teams are known to this day as old school run the ball, stop the run power football teams but that is onyl true of his first two championship teams, led by star runner Jim Taylor. Lombardi's last three Packers championship teams were in the bottom third of the NFL in both running and stopping the run, but at the top of the league in passing and stopping the pass. Modern football was here to stay at that point.

Modern football has a unique draw to American sports fans. Part is the system. High School and College teams draw on local and regional loyalties which for most Americans are an area of true passion, a feature of American culture people from other countries often struggle to understand. They create stars that go on to the NFL as established names with loyal followings. The best NFL teams feature this dynamic by combining multiple star players from multiple regions into one cohesive whole. The game itself is exceptionally appealing as well.

There are a lot of draws to NFL competition. The athletes are not only exceptional, they are exceptional in different ways. European football might feature athletes just as skilled, but they have similar body type and physical abilities, with the main difference being level of skill more than type. In the NFL you see enormously strong and powerful athletes, incredibly fast athletes, and athletes with exceptional agility and coordination all interacting on every play.

Before the play lines up you are presented with two formations, much like a chess problem which has to be evaluated and solved quickly. As the play initiates it does so with violent impact between linemen and sprinter starts by skill players. There is a lot of action all happening in a small space, a lot of visual input to analyze. The play itself is always a high risk high reward affair. Players are at real physical risk, and the plays can all turn very good or very bad instantly, both of which serve to heighten the sense of drama. You also have the NFL quarterback, a player who stands on the field calmly in the midst of all of this, is expected to successfully identify everything that is happening and take control of the situation by getting the ball to the right spot at the right time. No sport really combines these extreme elements of intellectual challenge, dramatic intensity from risk, and incredible athletic skill displayed in such a crushingly small space of time. When people are exposed to this at a high level, featuring players they already feel connected to, with high stakes on the line it's an addictive spectacle with an incredibly broad appeal. At least in the US, where it's a juggernaut.

The short answer is because thats what the cool kids were playing, ie college students. The long answer, I'm adopting from an earlier answer I did.

Rugby Football made its way to the USA in 1869, where it began to be played in a modified form by American college students, while Association Football, despite having been played a few years earlier for the first time in the US, wasn't latched onto in the same way. It is important to note that both those sports had only started to become codified themselves in the past decade, so were quite new.

At that first game in 1869, between Rutgers and Princeton, they were playing essentially rugby, but the rules evolved over time. The number one person responsible for this was Walter Camp of Yale, who pretty much wrote the rule book for american football.

Before Camp, each school had its own rules variations, some more like rugby as we know it, some closer to the modern (American) football game. Schools would also play against Canadian teams occasionally, usually McGill University. Generally speaking, you played by the rules of the host school, but this obviously was problomatic. Camp led the committee to agree on a set of rules for everyone, some times in the 1870s. He is responsible for the line of scrimmage, as well as the 11-man sides (originally it was as high as 25!).

The game was further changes in the early 20th century because so many people died playing it! President Roosevelt supposedly had to intervene and say "do something about this, or the game will be banned." This resulted in the introduction of the forward pass so make the players spread out more.

So anyways, at this point, we have a game that over 30+ years has very much changed from its Rugby origins, and is amazingly popular on college campuses. It long since expanded from its Ivy League (although the Ivy League didn't actually exist yet I should point out) origins and was being played nationally at schools both big and small. And it was popular. And it kept growing. East Coast to the Midwest and then to the West Coast and into the South. By the 1930s it was everywhere. But the college game was much more popular than the pro game, as it was a great way not just for schools to stoke a rivalry, but for the states as well via their big, public universities. The NFL, founded in the 20s, was originally just a regional thing in the mid-west, and many college stars didn't even go to the pros after.

It wasn't until the 1950s, and the introduction of television, that the NFL really took off. The 1958 NFL Championship game (this was before the AFL and the Superbowl) between New York and Baltimore is wildly credited as being the "birth of the modern NFL" since it was one of the first nationally televised pro football games. And from there is just kept going. Hope that helps!

Sources on this: Some is just stuff I've picked up over time, but primarily I'm drawing on three books I have read on this topic (I am doing it off the top of my head since none are handy at this moment. Any errors are to my memory, not the author). The Big Scrum by John Miller; The Glory Game by Frank Gifford; and The Best Game ever by Mark Bowden.

Now, since you asked for other sports as well, here is an old answer I did for Baseball!

In the case of Baseball, its rise is the case of a small, mostly regional game enjoying a quick, rapid rise in popularity thats to a national event that brought it to the attention of people all over the country, namely the Civil War.

Bat and ball games had been around for ages, being played in Great Britain for centuries, and they no doubt came to America from there. Cricket, baseball and rounders all have their origins somewhere back there.

Now, with baseball specifically we start to see regional variations that kind of, sort of resemble baseball being played in America in the first half of the 19th century. The Knickerbocker rules - which date to 1845 - were the dominant set of rules being played in the area of New York, but they were by no means the only set of rules. The game would be kind of recognizable to a modern observer, but rules such as making an out on a catch after one bounce would be pretty weird now. It also lacked 'balls', although three strikes was a rule.

Now, as I said, the Civil War was a key component of the rise of baseball. When the war started, hundreds of thousands of Union soldiers from all other the country found themselves camped together and with a lot of downtime. Baseball became an exceptionally popular game for them, and with New Yorkers everywhere, the Knickerbocker rules found themselves to be the most popular used (as opposed to the Massachusetts rules popular in New England, and Cricket, which was played in the Philly area), essentially becoming the de facto rules for Union Army baseball. When the war ended and the Army disbanded most of its men, soldiers headed home, bringing with them either a new game, or else a new set of rules for one they had already played.

In the wake of the war, amateur athletic clubs started sprouting up all over, and within only a few years, the first Professional club was founded in Cincinnati, in 1869. More clubs followed, and in 1876, the National League was founded (It was not the first league though, as there had been a pre-war association, the NABBP, that governed amateur clubs in the New York Area). The rules continued to evolve beyond that point - here is a handy list - but by 1889 or so (with the introduction of 'four balls'. It had been as high as nine in the past) I would say the game very much resembled what we have today.

So there you go, as with most games (Basketball is the only one I can think of off hand that was essentially created out of thin air), it was a slow, evolutionary process from earlier ones. That being said, the most important point in its development was the American Civil War, which not only helped to speed up standardization of the rules, but created a nation wide network of fans and players.