Yes, it's true. The earliest ancestors of indigenous Americans hunted large mammals, horses included, but they never domesticated them, and horses had died out in the Americas by roughly 11,000 years ago. So for the vast majority of human expansion in North and South America, horses didn't exist. Indigenous American cultures didn't have domesticated draft animals except for dogs (which dragged sleds and travois) and, in the Andes, llamas. None of these were suitable for riding.

Domestic horses were one of the new technologies Europeans brought to the Americas, and Natives quickly saw the advantages. Raising, training and riding horses is complex, and the earliest Native horsemen learned this from working for Spanish colonists. For example, Santa Fe was founded by the Spanish in northern New Mexico in 1610, and by the 1650s the Spanish governor was already complaining of attacks by mounted Navajo-Apaches. Within another few decades, the knowledge of horsemanship had spread north from the Midwest Plains to the Pacific coast, and by the early 1700s, you had previously sedentary farming peoples radically changing their lifestyles as they became the "horse cultures" of the Plains and the West.