First off, most NFL teams let a local high school or college marching band do a half-time show. In most cases, the school that provides the marching band rotates from game to game.
The Buffalo Bills always use the Attica High School Marching Band and have had an exclusive contract with that high school, since they started to play their home games at Orchard Park NY. The Baltimore Colts had an all volunteer marching band, but when they moved to Indianapolis, in 1984, the band did not follow. When the Cleveland Browns moved to Baltimore, the band became the Baltimore Marching Ravens and remains an all volunteer band, with no school affiliation.
Far and away the most famous marching band in the NFL is the Redskins marching band. They are also an all volunteer band and have been performing since 1937. They were the brain-child of George Preston Marshall. His wife, Corine Marshall, wrote the Redskins fight song, which the band plays at every home game. During the 1950s and 1960s, the Redskins even took their band along to road games against the New York football Giants.
Source: http://www.redskins.com/team/history/marching-band.html Note: the author of the piece on the Washington Redskins Marching Band, does not consider the Attica High School Band to be an official NFL marching band

It is my understanding from military history I received while in the service that the first college marching bands were from the military academies and schools with large ROTC programs. The idea of a brass and percussion-heavy band comes from the drummers you often see marching in front of a division of soldiers, and later the bugler. Also, marching in a coordinated manner and playing music is an excellent way to train cohesiveness in large groups of people. These shows were introduced to a football environment which other schools then picked up. The NFL has never been a stepping stone to military service, so it is not needed. However, at some games a local college band will occasionally perform, but it isnt often televised.