While the following videos of vintage hay presses are not the exact same as Dwight Yaeger's Kansas City Lightning Hay Press, they at least give an idea of how hay presses worked.
A mid-1910s McCormick-Deering hay press, powered by a 1915 6 hp Stover/Sandwich (double tagged), working at the Antique Tractor & Gas Engine show at Petersburg, Neb.
An International Harvester Hay Press No. 15 powered by a 1951 Farmall Cub.
An International Harvester Power Press powered by a Farmall M.