First “Big Apple” explanation: February 18, 1924
Above, the header from the 1924 newspaper column of John J. Fitz Gerald. Click to see a portion of the column which includes his use of "Big Apple." Part of a 1926 column is also available.
I found the first John J. Fitz Gerald "Big Apple" explanation. An apple graphic is featured in the column head. From right to left on the apple's New York City skyline appear to be the Woolworth Buidling (then the world's tallest building), the Park Row Building and the Municipal Building.
The column's text is now famous. From the New York Morning Telegraph, "AROUND THE BIG APPLE" with John J. Fitz Gerald, February 18, 1924:
The Big Apple. The dream of every lad that ever threw a leg over a thoroughbred and the goal of all horsemen. There's only one Big Apple. That's New York.
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Two dusky stable hands were leading a pair of thoroughbreds around the "cooling rings" of adjoining stables at the Fair Grounds in New Orleans and engaging in desultory conversation.
"Where y'all goin' from here?" queried one.
"From here we're headin' for The Big Apple," proudly replied the other.
"Well, you'd better fatten up them skinners or all you'll get from the apple will be the core," was the quick rejoinder.