Why jazz?

Because it resists easy classification, jazz can be infuriating to a society that has grown accustomed to genre-based, formulaic music marketing. Even jazz musicians squabble over what jazz actually "is." For the sake of sincerity I'll give you a straight-ahead definition (as opposed to a swung one):

Jazz is the product of several converging musical traditions, primarily those of African slaves and their ancestors. The musical syntax of Europeans was successfully incorporated into slave musical traditions through a process known as acculturation. Other cultural influences (notably those of the Caribbean and Latin America) abound in jazz.

Jazz is typically distinguished from other genres by its emphasis on improvisation; i.e. music that is produced by the performer at the moment of performance. Improvisation has been called "spontaneous composition," since the act of composition is taking place in a performance setting. This defining trait can take decades to master and is not merely "making stuff up," nor does it imply a lack of preparation. Successful improvisers spend years immersing themselves in the components of jazz harmony and theory. You have to build the dam before you can uncork it.

Another characteristic of jazz is its cyclical nature, in which sequences of musical materials are repeated. These cycles provide a backdrop for a soloist's improvisations. The soloist can then choose the parts of his/her improvisational "bag of tricks" that are best suited for a given cycle. Jazz possesses an innate rhythmic fluidity, a feeling of pushing and pulling, that is often called "swing." This is a stylistic trait that emphasizes certain pulses and beats over others. Contrast this with a European "classical" performance style, where all beats are given equal emphasis.

Or you can just take it from Sir Duke:

"Put it this way: Jazz is a good barometer of freedom… In its beginnings, the United States of America spawned certain ideals of freedom and independence through which, eventually, jazz was evolved, and the music is so free that many people say it is the only unhampered, unhindered expression of complete freedom yet produced in this country."
- Duke Ellington (Source)