It's part of the american folk mythology (for a real obvious-in-hindsight example, consider Star Wars (original trilogy): Luke goes from farm hand to ace fighter pilot with no training, and becomes a passable jedi with a few months of training...).

It probably originates 50% from the urban/rural divide and 50% from the underlying outlook of protestantism. In brief:

- much of american history consists of pioneers going and settling the land, then -- decades later -- the 'experts' show up (these could be anything from farm scientists to law enforcement to general government busybodies). So there's this weird split between respect for learning and accomplishment (no, really!) but on the other hand the belief that you can really boostrap yourself / figure things out by yourself just fine. Anything that plays into the belief you could perform at an 'expert' level with only a very minimal amount of training is therefore good entertainment.

- the basic protestant outlook is that your (personal) salvation is predetermined (though perhaps you can mess it up), and that consequently nothing you do can earn salvation if you weren't already gifted it; however, there's a parallel belief that God will show favor upon the saved, eg by granting them success in their worldly endeavors...and so, loosely speaking, a protestant-minded person would interpret someone attaining expert-level performance in a very short period of time as a sign that that person had been gifted with salvation.

At this point in time I'd say neither of those reasons is self-consciously driving the meme's dominance, but those are some of the deep roots of that particular outlook, most likely.