Anniversary Day was created by the New York State Legislature in 1959 in commemoration of the organization of Sunday schools. This was originally connected to "Rally Day" Parades held by Sunday Schools in Brooklyn, dating back to at least 1838.

In 1925, 100,000 people marched in the Brooklyn-Queens Day parade.

More Info: NY State Consolidated Law, Article 52, Section 2586:

S 2586. Anniversary day as a holiday in the public schools of the borough of Brooklyn and in the borough of Queens, city of New York. The first Thursday in June in each year, except in those years when the first Thursday in June occurs in the same week with Memorial day, and in such years the second Thursday in June, known as anniversary day, and celebrated in commemoration of the organization of Sunday schools, is hereby made and declared to be a holiday in all the public schools in the borough of Brooklyn and in the borough of Queens, city of New York, and the board of education of such city is hereby authorized and directed to cause all the public schools in such boroughs to be closed on such day.

Now all NYC public schools are closed on "Brooklyn-Queens Day" for a Chancellor's Conference Day.