It didn't. The accent spoken in Oklahoma, for example, is nothing like what you'll hear in the Georgia and Carolina Piedmont, and it's also quite foreign to a Coastal Southern speaker. Rhotacism is a big thing that separated them until recently (the southern dialects were non-rhotic, what you describe elsewhere was).

Essentially, you perceive these dialects to be similar due to similar connotations and social standing, as well as unfamiliarity.

Now, to backtrack a bit, they do share a few features, but those are simply because of the expanse of those feature (such as the pin-pen merger) in recent times.

Compare Gone With The Wind: https://www.youtube.com/watch?v=jlBc_nlA2so

With Woody Guthrie (Oklahoma): https://www.youtube.com/watch?v=PlgXghaJCK0 (starts at about 2:00)

What this suggests is that the Western dialects have a lot more in common with the Appalachian dialects (strong rhotacism and r-insertion being big features) and other frontier forms of speech. It's a lot different than what you'll find in the Piedmont and Coastal regions.