Jefferson scholars are not selected from the general application pool, they are nominated by their HS. The Jefferson Scholars Foundation is run by alumni, not by the university. http://www.jeffersonscholars.org/
Over 4200 schools are eligible to nominate one student each, but they don’t all nominate someone every year. In the 2013-2014 cycle, 34 Jefferson Scholars were selected from among 1750 nominees. That means that even if you’re good enough to get your school’s nomination, you still have only about a 2% chance of actually winning a J. scholarship. It’s much harder than getting in to Harvard or Stanford, for example.