Background
After defeating the Union once again at Gettysburg, Lee's army laid siege to Philadelphia for nearly a year, retreating only when Sherman's forces returned to the North. By this time, the people in the Union states had grown weary of the conflict. George McClellan was elected to replace President Lincoln, and in April 1865, a peace treaty was signed with the Confederate States of America.
However, the Emancipation Proclamation had made an impact in the Confederacy, and slavery was fraying at the edges. By 1890, slavery was abolished in the South, but all the freed slaves were expelled, mostly to the USA. After the turn of the century, relations between the CSA and the USA were normalized, and people (except blacks) and commerce flowed freely between the two nations.
Jackie Robinson's debut for Brooklyn in 1947 angered some Southerners very much. A few wealthy Southern gentlemen thought there ought to be a professional baseball league that "folks could enjoy without the disturbing presence of coloreds and foreigners." The following year, the Confederate Baseball Association was born, with clubs in Atlanta, Charlotte, Dallas, Houston, Jacksonville, Memphis, Miami, Nashville, New Orleans, Norfolk, San Antonio, and Tampa. (In a grudging concession to the on-field product, the owners agreed to allow players not just from the CSA, but the USA as well.)
[Editor's Note: Please don't take this backstory as evidence of prejudice on my part. I only came up with this premise to explain (a) how a major league could develop in a small enough area to support only 12 teams and (b) why all the players would have American names (c/o Malleus Dei) and origins. If anything, I probably thought about this too much, and thus it would help if you, the reader, didn't think much about it. 
The short version is: the South won, but now we get along, and it's pretty much like Canada except warmer and more exclusive.]