There were two main things that attracted settlers to the Great Plains in the late 19th century.
The first of these things was the lure of large amounts of relatively cheap land that could be cultivated. The government helped with this by enacting the Homestead Act of 1862, which gave settlers free land. The land was more able to be cultivated than it had once been because of the invention of new technology such as better plows.
A second factor that attracted settlers was the presence of railroads. The railroads offered a way to get to the plains and get crops back to population centers. The railroad companies helped this process along by doing a lot of advertising to lure people to the Great Plains (because the railroads owned much of the best land and because a rising population meant more business for them).
So, the Great Plains became more reachable, more affordable, and easier to cultivate in the late 1800s. This is much of why people moved there.
It is also important to mention that government efforts to remove Indians from this area made settlers much more likely to move to the area.