My mistake. The law authorizing central school districts was passed in 1914, not 1925. In 1925 the state offered financial incentives to spur the formation of central school districts, including 50% of transportation costs and 25% of construction costs. (The same bill, the Cole-Rice Law, also created equalization aid to bring per-student spending up to minimum statewide standards.) The centralization process picked up in the 1920s and 1930s, and it continued through the 1960s.

Central school districts were an effort to consolidate the old one-room school houses in rural areas (each school corresponding to a "common school district") and merge them with nearby village schools. For example, the rural Town of Sodus, in Wayne County, had about 23 school districts, with a total of 27 teachers, at the end of the 19th century. Creating central school districts required new buildings and busing, but it allowed the former districts to pool their resources and offer secondary education.

Apparently, there is no thorough history of education in New York, but see:

James D. Foltz, "History of the University of the State of New York and the State Education Department, 1784-1996."