People get this view primarily by using the information of family trees in the Bible and tracing the lineage back to Adam. It comes from several places in different books. The most notable person to constructive a chronology of the world by doing this (although he doesn't base it solely on that) was James Ussher. It was included in annotations in several popular versions of the Bible from about 1700 onwards. Becuase of this, his chronology, or at least his methodology of using the biblical ages, approximate dates using the Jewish calendar, etc., became very popular in the U.S.

As the article on him notes, even before the common acceptance of the theory of evolution as likely valid and well before the suggestion and proof of the big bang theory, theologians disputed his method and claims. They were doing this even in the eighteenth-century.

For what it's worth, youth earth creationism is a minority view within Christianity. It is so well known in popular culture because some of its proponents are quite vocal (and probably because it's fun to make fun of...). I have two degrees in theology, one from a very conservative, charismatic Christian school and the other from a moderate, mainline, nationally known university. At neither place did I find a professor who believed the earth was young, whether it was one who taught in the sciences, biblical studies, or theology.

edit: grammar