It doesn't really mean anything. It's a kid's joke/prank/game sort of thing. Fairly nonsensical. When a kid finishes an apple, he shows the core to his friends and says "Apple core." From there, the "Baltimore" is there only for the sake of rhyme. (Other versions use "nevermore" or "say no more." In my school, we used "tell me more.")
Then the "apple core" kid asks, "who's your friend?" To which the one who was the first to say "Baltimore" says the name of another kid in the group.
Now, from here some versions of the game are longer than others; employing more or fewer "apple core" rhymes (we used a short version). But, long story short, the apple core is thrown at whoever was named as the "friend." And the kid who started it by saying "apple core" (and threw the apple core), finishes the game by saying, "not no more."
It's really just a kid's useless rhyme like "hey diddle diddle" combined with the opportunity to chuck used food at someone in a socially acceptable manner.
Actually, by high school, my friends and I had abandoned the apple altogether. Someone just randomly said "apple core" and whoever said the rhyme then picked some one else as the "friend." And the friend got punched. Nifty, eh? What's better than a little violence among friends? Rhyming Violence, that's what.