The short answer is: because baseball became so popular.
If you look through newspaper archives of the larger U.S. East Coast newspapers before the Civil War, articles about cricket matches are pretty common. Here is one from the May 24, 1854, edition of the New York Tribune describing the first cricket match of the 1854 season in the New York metro area.
But back then, sports were amateur, and while cricket was well known and relatively popular in North America at the time, it had a formal set of rules with a set number of players. So when playing a pick-up game in less than formal conditions with a handful of players, any of the other bat-and-ball alternatives might be played. These other games weren't really based on cricket--by the early 19th Century, there had simply developed regional "house rules" for bat-and-ball games. In New York City, the preferred bat-and-ball game was called baseball, while up in Boston, they played the "Massachusetts game", and down in Philadelphia and the Pennsylvania countryside, they played "townball".
Really, they were all based around the same concept of hitting a ball with a bat and touching bases to score. It was a lot more like, "OK, this stump will be the home base, and that tree will be the other base, and we each bat in turn until one team scores ten." There were regional preferences for rules of such games, but kids would know and would have played all sorts of variations of hitting a ball with a stick and tagging a base, and none of them were very formal, except for cricket. So cricket would be played, even in a pick-up game, but because of the circumstances of playing such games, the rules would be adhered to as loosely or as strictly as appropriate.
By the 1840s, English immigrants in America started forming cricket clubs, and were popular enough that they would even sometimes pay instructors a salary. Additionally, they often excluded non-Englishmen from joining these clubs. At the same time, in New York City, baseball took off in a big way. The most direct reason for this is that a ball club called the Knickerbocker Club formally wrote down the rules they were using, and they began challenging other clubs around town with these rules, and they published those rules in 1848 in a cheap pamphlet, and this version of the game caught on. They were the first formal rules of baseball.
When the Knickerbocker Club had started, they were playing all sorts of bat-and-ball games. The club was made up of amateur New Yorkers, many working for the fire department, but also included businessmen and lawyers and men from all walks of life. The president of the club was a guy named Duncan F. Curry, who worked in the insurance industry, and described the early days of the club this way:
"For several years it had been our habit to casually assemble on a plot of ground that is now known as Twenty-seventh street and Fourth avenue, where the Harlem Railroad Depot afterward stood. We would take our bats and balls with us and play any sort of game. We had no name in particular for it. Sometimes we batted the ball to one another or sometimes we played one o'cat."
He then went on to say that one of the other club members, Alexander Cartwright, came to the playing field one day with the idea of playing nine innings, with each half inning made up of three outs, and the two teams would switch off between fielding and batting. Cartwright also suggested the diamond shape for how the bases should be laid out. Said Duncan:
"At that time, none of us had any experience in that style of play and as there were no rules for playing the game, we had to do the best we could under the circumstances, and aided by Cartwright's judgement."
From these modest origins, the club formalized a set of rules around this style of bat-and-ball game, and these rules came to be known appropriately as the "Knickerbocker Rules". The game became known as "baseball" because that was one of the names floating around for bat-and-ball games in New York City at the time, though before that, "baseball" could be played any which way.
And even under the Knickerbocker Rules, there were still some key differences that would be changed later on--those original rules stipulated the ball had to be thrown underhand ("pitched"), and that the ball could be caught on the first bounce for an out, among other differences.
Even then, the Knickerbocker club and the clubs they would challenge weren't strictly playing baseball only. They would often change it up by playing cricket, or "old cat", or whatever bat-and-ball game suited them when they showed up to play.
And other clubs had their own preferred rules. This led to several of the higher profile ball clubs in New York City organizing a convention in 1858 that led to the formation of the National Association of Base Ball Players. Though it called itself "national", originally, all the clubs were in metro New York. The association standardized the rules for their member clubs, published them annually, and organized matches between its members. Any club was invited to join for a fee, and the organization grew throughout the East Coast slowly at first, but then exploded after the Civil War. Baseball became so popular that, after the NABBP convention in the 1870-71 offseason, the organization dissolved, with some of the higher profile clubs (by then able to attract paying spectators) forming the National Association of Professional Base Ball Players, while the amateurs split into regional governing bodies.
Through all this, many of baseball's early proponents were immigrants, many from England, and many with a cricket background. Among them were Henry Chadwick, born in Exeter, Devon, who spent a decade as a cricket reporter for the New York Times before baseball fever caught on, and he began enthusiastically covering more baseball than cricket, to the point where he was covering baseball exclusively. Chadwick would become the first journalist inducted into the Baseball Hall of Fame.
Another English-born New Yorker influential to the early game was Harry Wright, who was born in Sheffield, South Yorkshire. He had immigrated to New York with his parents and siblings when he was just a boy. His father, Samuel Wright Sr., helped form the professional St. George's Cricket Club in New York (with a playing field in Hoboken) after immigrating. This cricket club was one of the most prominent in the city and, fatefully, it barred Americans from joining the club, thus limiting the competition and spread of that game. Even the more American-friendly cricket clubs often limited membership by charging higher dues than other ball clubs in the city.
The elder Wright taught his sons, Harry and George, how to play cricket when they were kids, and Harry played it professionally, joining the Union Cricket Club in Cincinnati, Ohio. He then supplemented his income by getting hired on as the "manager" of the area Cincinnati Base Ball Club ("manager" in this era would have been more like a "golf pro" at the local country club today). In 1869, the Cincinnati club hired a bunch of other baseball players on salary, mostly from back East, and toured the country as the Cincinnati Red Stockings, the first officially recognized professional baseball club, with Harry Wright effectively leading the team.
Younger brother George Wright was also a member of the Red Stockings, and perhaps their best player. When the club folded once the National Association of Professional Base Ball Players was formed in 1871, George left to play in that league's inaugural season for the Boston club that is now the Atlanta Braves. He played for the team for eleven seasons, the last six under the National Association's successor, the National League (which today makes up one of the two leagues of Major League Baseball). Both George and Harry Wright are now members of the Baseball Hall of Fame.
Really, even by the time that Harry Wright left New York to play cricket in Cincinnati, cricket was already dying. A good summary of why this happened is contained in the chapter "Why Not Cricket?" in the book Baseball’s First Inning: A History of the National Pastime Through the Civil War by William J. Ryczek. As cricket became more insular during the 1850s and 1860s, the more welcoming baseball clubs became popular New York area attractions as a spectator sport. Among the most prominent clubs in the 1850s and 1860s included the Eckford club, the Atlantic club, and the Excelsior club, all located in Brooklyn; the Union club in the Bronx; and the Mutual club, the Gotham club, and the Knickerbocker club in New York City (Manhattan). Some of these clubs would stay amateur or else folded when the NAPBBP took the game pro. The Atlantics, the Eckfords, and the Mutuals all went pro in 1871 as charter members of the new league, competing against pro and semi-pro teams all along the East Coast.
Author Ryczek quotes a September 1863 New York Times article that was indicative to how the popularity of cricket and baseball were then diverging:
"In September 1863, the Atlantics and the Mutuals played a baseball game on the St. George cricket ground in Hoboken. '[T]here were more persons present in the occasion,' reported the Times, 'than there have been the entire season on the St. George, New-York, and Manhattan cricket grounds combined. This shows that cricket has been almost obliterated by base ball, which, but ten years since, was in its infancy, and could only count a corporal's guard to witness a match, while a cricket match would be attended for a couple of days together by from 8000 to 10,000 persons.'"
(...cont'd...)