Americans started to move out west during the 1800s because of Westward Expansion, which meant a lot more land, resources, and jobs were available.
Westward Expansion started with the Louisiana Purchase, which was when France sold a good portion of land to the United States. With the Louisiana Purchase, the United States had now doubled their land. Thanks to that, this meant a ton of new land was available, along with resources and jobs. With the Gold Rush, a lot of people moved west to simply try to get rich quickly by finding gold.
The belief in Manifest Destiny also caused a lot of Americans to move out west in the 1800s. Manifest Destiny is the idea that Americans have the God-given job to expand over the United States, even spreading their beliefs with them. This made a lot of people move west, as they felt it was their purpose to help expand the United States.