Why were horses so important to the mongols during the war

They were skilled with archery on horseback that is why they demanded horses for war.

Other than that, they serve as riding animals, both for the daily work of the nomads and in horse racing. Mongol horses were a key factor during the 13th century conquest of the Mongol Empire ... Men do the herding, racing and make the tack. Traditionally, men (or in modern times, women) also milk the mares.