hay means you can use horses to pull carts, carry soldiers, operate mills and pumps, mine coal, facilitate industrialisation, mechanisation, mass warfare, tax collecting, cities etc
since they have not got to spend all their time eating grass
Yup - JohnD has got it. Hay allowed humans to keep far more animals alive over the winter than before, and allowed them to be used to do work, and to efficiently provide food for more people.
It was the increasingly successful exploitation of animals that gave mankind not only the physical resources to develop and eventually industrialise, but also allowed them to move on from hunter-gatherer societies to larger more structured ones in which more and more people had the time to devote efforts to development and innovation rather than basic survival.
Without hay we would still be nomadic hunter-gatherers.