How is grazing land the cause of deforestation?

Grazing land or grassland is actually the result of deforestation, not the cause. The cause of deforestation is fire and defoliation.

Deforestation is actually nothing new to the natural world, as elephants are well known to push down trees and clear areas--unintentionally, of course, as they bring down the trees so they can eat the foliage better--within or of forested land, and this is essentially converted to grassland or meadows for other grazers.

The grasslands of North America have largely been helped by the First Nations some thousands of years ago with their use of fire to clear woody plants and encourage more land to be available for the millions upon millions of natural grazing animals that populated the landscape long before the first white settlers made a move to "tame the land." This so that they had more opportunity to hunt, but also as a means to wage war on rival tribes.