Quick answer: cheerleading originated in the United States in the late 1800s as organised college crowd-leading, and grew over the following century into a nationwide school tradition and, later, a distinct competitive sport with its own gyms, governing bodies and championship events.
Origins in College Football
Cheerleading's roots in the United States are tied closely to college football, where organised "yell leading" developed to coordinate crowd support at games - from Princeton's group yells in the 1870s and 1880s to Johnny Campbell leading the crowd at the University of Minnesota in 1898 - see When Did Cheerleading Begin? for the broader timeline.
Spread to High Schools
As cheerleading became established at the college level, the tradition spread to American high schools through the early-to-mid 20th century, becoming a common feature of school sports culture nationwide, particularly around football and basketball.
Growth of Training and Camps
Dedicated organisations and training camps - most notably Lawrence Herkimer's National Cheerleaders Association, founded in 1948, and Jeff Webb's Universal Cheerleaders Association, founded in 1974 - played a major role in standardising cheer technique across schools throughout the country, and in building cheerleading into a recognised American youth activity.
The Rise of Competitive and All Star Cheer
From the second half of the 20th century, competitive cheer grew alongside sideline cheer, helped by televised championships from 1978 onward, and culminating in the rise of All Star gyms from the 1980s - private, school-independent teams that now represent a major part of the American competitive cheer landscape. The U.S. All Star Federation was founded in 2003 to set their rules. See History of All Star Cheer for more.
Cheerleading in American Culture
Beyond sport, cheerleading has become a broader cultural touchstone in the United States, closely associated with school spirit, and has been depicted extensively across American film, television and media over the decades.
A Note on Dates
Some early details vary slightly between sources. Where sources disagree, we give the most widely cited version and avoid false precision - the sources we used are listed below.
Sources: Cheerleading (Wikipedia); Being a Cheerleader - History of Cheerleading (Varsity); Varsity Spirit (Wikipedia); U.S. All Star Federation (Wikipedia)