Well, classical flutists and violinists have been using vibrato for centuries.
In many African languages (I am told) pitch conveys meaning so timbre is often used to convey emphasis. This carries over from speaking to singing.
When Africans came in contact with European musical instruments and traditions, the use of vibrato in the European tradition was therefore similar and familiar to their experience in singing and speaking; early mentions of slave musicians in America often mention their use of vibrato and the tonal quality of their playing as well as their rhythmic usages. The creators of jazz were also in that tradition; they generally had extensive experience with European musical traditions through brass bands, piano music, etc., and so vibrato was naturally carried over into jazz.
The most characteristic aspect of jazz vibrato usage is what's called the "terminal vibrato" where a note starts straight and vibrato is applied after the start of the note, often increasing in width (and sometimes in speed) as the note persists. The application of the vibrato can create an effect much like a second breath accent thus giving a sforzando effect. In jazz playing you don't just play a note and let it lie there, you do something with a long note.
A tradition in jazz (though, sadly, disappearing) that does not exist in classical music is the concept of each musician having his "own sound". In the jazz tradition this is probably the most important aspect of musical development, more important than technique, or theory, or sightreading, or knowing hundreds of tunes. No one wants to be told "you just sound like a copy of XYZ." There are many things that go into a musician's "own sound" and his individual use of vibrato is one of those - an important one. I once was recognized from across a football stadium when warming up by sound alone.