The above quote appears in Frederick Turner's highly influential essay "The Significance of the Frontier in American History." Turner published the essay three years after the US Census Bureau declared that there was no longer any "frontier" line or "wilderness" awaiting white settlement.
Turner argued that the continuous existence of a frontier, a new, unsettled (by whites) area of land waiting to be tamed, had a profound influence on both the American character and American society—and in turn, Europe. In the quoted passage, Turner asserts that the frontier was important because it promoted democracy and individualism. Because there was no preexisting social structure on the frontier (we have to keep in mind that Turner did not value Native societies), there was very little outside force to control settlers' behavior. Settlers came in essentially as family units, with no other social structures in place, and got used to making their own decisions, treating other people they met as equals, and doing their own thing. This led to the American Revolution and a democratic system because frontier people grew unused to bowing to central authority and began to perceive this form of governance as repressive and restrictive. When Turner writes that the frontier produces "antipathy to control and especially to direct control," he means that people grew so used to being unsupervised and allowed to run their own lives that they grew to dislike any interference.
The frontier provided an outlet in which the most individualistic people in society could find space and freedom, and it taught people to value individualism and democracy. Because of it, the right to direct one's own life and the right to participate in governance became enshrined as among the most important American values—and provided a model for European democracy as well. As Turner puts it:
The effect reached back from the frontier and affected profoundly the Atlantic coast and even the Old World.