One element is the willingness to believe that America/the United States is exceptional in the world/historical scheme of things, and that the Christian God intended it to be this way.
Some Christians believe that America was "hidden" by God in order to provide a refuge for religious dissenters to escape persecution in England; to provide a "beacon" of proper submission to God as an example for England and the world, via Winthrop's "City on a Hill." A strong strain of this thinking still exists among some American Christian believers.
Add to this an eschatology, popular among evangelicals, which assigns a specific role to the United States and Israel.
Add in an interpretation of scripture which convinces them that a nation that "forgets God" will be punished (especially a "Redeemer Nation" as they believe the United States to be, ala ancient Israel).
Combine this stuff, bake it in the climate of fear and uncertainty which often accompanies socio-cultural-economic-political change (both national and international) and you get a theocratic-minded push to make it a test for national leadership, and a standard for national culture and law.
It has become especially obvious to us since the rise of the Religious Right, but it has been a strain of thought with us since the foundation of the republic (although modified by the introduction of Dispensational theology, etc., over the years.)
It's a very, very dangerous phenomenon.
Edit: added "cultural" in the fifth segment.
