This question is a bit general, so if there’s some more specific element you’re looking for please let me know and I can focus on it. This answer is a bit broad-strokes, so wherever you want to zoom in just say the word.

Marching bands in general stem from military bands, and the long tradition of instruments used on the battlefield - see fife and drum corps, for example, though this goes back to antiquity. Hence also the uniforms, which are also military dress inspired though have become somewhat derivative since. So why the military-style unit oncollege campuses? This is because they stemmed from the ROTC programs at those schools.

The songs you speak of are considered “fight songs.” They are written as marches (a la John Philip Sousa) and are usually - though not always - written by an alumnus of years gone by. Some use original melodies, others borrow from classical music. I believe the oldest (according to the school website) is “For Boston,” from Boston College.

There is unfortunately not a lot of formal literature published about the development of modern marching bands. It is a fairly niche area and its status as entertainment combined with a disdainful view of the activity by many serious academic musicians likely contributes to this. If you want to know more about the history you may want to start by researching military bands. It would also be worth looking at music education pioneers like Lowell Mason, and the earliest college bands at schools like Notre Dame.

However, this is an interesting resource with valid information: https://www.lipscomb.edu/windbandhistory/rhodeswindband_09_americanschoolband.htm