By the late 1800s, the development and popularity of the brass band had been established. These groups of roughly 4-20 members consisted of percussionists (cymbals, snare drums, triangle), brass instruments (trumpet, trombone, sousaphone, baritone), and woodwind players (flutes, piccolos). Almost every community in America had a brass band that performed for everyday events as well as formal ones such as dances. Imagine walking down the street, and seeing a group of instruments just going at it while you went to pick of groceries.
Because segregation was still an issue around this time, each community, assuming they had enough members to support each, had a black and white brass band. These groups were important to black people because they were of the first fraternal groups blacks could be a part of.
Not to give too much credit to a single person, but John Phillips Sousa was the one to write a vast majority of brass band music. Some of these songs have been incorporated heavily into our lives today (Stars and Stripes Forever for example).
A good example of a modern day brass band is the Hypnotic Brass Ensemble. The group is much smaller than most of the older brass bands, but they retain the same brass band elements. The bass lines are on the sousaphone, trumpet gets the melody, and percussionists hold down the beat.
An important thing to note about this time is the development of the instruments themselves. Trumpets and other valve instruments are becoming much more refined as musicians are demanding more and more out of particular instruments. These developments are crucial to how modern instrumentalists get their particular tones.
"Men have died for this music. You can’t get more serious than that." - Dizzy Gillespie
I like band music a lot. But I'm not sure how John Phillips Sousa's band music turned into jazz. Here's a marching band that is clearly jazz..
It's not necessarily that Sousa's music contributed directly to jazz but more so in that it brought popularity to people going out and seeing music. On top of this, it also got more and more people interested in harmonizing horn lines. Over time, this development turned into more modern harmonization that we hear today.