So, any answer to this is gonna be a gross oversimplification, but here goes.

Black antebellum music largely falls into three categories: gospel, work songs or chants, and the murky beginnings of what we refer to as "old-time music."

Gospel is the easiest one, there was a strong religious tradition in the South, and preachers both black and white preserved and transmitted songs throughout the region.

There's a dearth of adequate primary sources, but there's evidence of black string bands who provided entertainment at white parties. They traveled, and presumably learned the way all musicians learn, teaching each other songs and techniques.

Finally, one could speculate that songs traveled much the way they traveled elsewhere, just slower. You learn a song from a slave the next plantation over, you teach it to your cousin, who teaches it to his daughter, who lives in town with your master's son, who teaches it to a friend, who gets sold down the river and teaches to to someone in Louisiana, and so on and so forth.