Part I:

Taking a look at U.S. census data, we can start to form a picture which might help explain what appealed to people:

Decade	1970	1960	1950	1940	1930	1920	1910	1900	1890
Urban population as a percentage	73.6%	69.9%	64.0%	56.5%	56.1%	51.2%	45.6%	39.6%	35.1%

From 1940 to 1950, we see the single biggest jump in concentration from rural areas to urban areas. This indeed makes quite a bit of sense: not only did America expand industrially during WWII, but saw increased infrastructure projects and the suburbanization of America after WWII. On top of that, Europe had destroyed most of its manufacturing during the war and European emigration decreased. Manufacturing is naturally located in urban areas and farming underwent continued industrialization, making population flow towards cities inevitable. As we will see, there was significant population transfer from mountainous regions outward.

I'm going to quite extensively from "The Hillbillies in the Living Room" by Anthony Harkins. This 30 page text will pretty much entirely answer your question, so if you're interested in a much deeper discussion I'd suggest you read it. I'm abridging it extensively here:

Appalachian outmigration began in earnest around the turn of the century, grew significantly during the Depression years, and then leapt upward in the early 1940s in response to the boom in industrial war production work during World War II.

Stereotypes of the population transfers were not positive, however. In fact, the people from the Appalachian regions were looked down upon as kind of a scourge:

Free from the institutionalized racism faced by their Southern black counterparts, the majority of Appalachian migrants were able to find employment and slowly to improve their economic situation. Yet the perception of Southern white migrants as an inassimilable and unwanted population was widespread in the postwar Midwest. Job listings that "No Southerners need apply" and restaurant owners who refused to serve "hillbillies" were not uncommon...

A growing regional concern in Midwestern cities since at least the 1930s, the Southern migrant "problem" was first presented to a national audience beginning in the late 1950s. Striking to present day readers is the explicitly racialized depiction of this population. James Maxwell's "Down from the Hills and into the Slums" (1956), for example, opens by quoting an Indianapolis resident fearful of what she considered an uncivilized and dangerously independent population. Maxwell then informs what he presumes to be his shocked readers that this group was not "Puerto Ricans" or "Mexicans" but "white Anglo-Saxon Protestants," a group "usually considered to be the most favored in American society." Heavily reliant on antagonistic law enforcement officials for informants, he quotes authoritatively a police officer who accused the newcomers of a wide variety of crimes including "shootings, child neglect, rape ... [and] incest..." Although he stresses the difficulties migrants face, Maxwell generally portrays them as a backward people badly out of sync with urban ways and mores.

Suffice it to say, there were tensions with these population flows - tensions which worked their way into popular culture. However in television the stereotypes ranged, they were not entirely hostile. As depicted in The Real McCoys:

[One episode] also explores the limits of such progress and the sort of Faustian bargain the McCoys' relatives have made in sacrificing a traditional rural ethic for mass consumer goods. In the subsequent episode, after all of Grampa's relatives are fired when he refuses to sell to the box company a key plot of land that Great Grandma wishes to hold onto for nostalgic reasons, Grampa tries to comfort his kin by assuring them that "McCoys are people o' the soil" and that they can all go back to farming. Yet to Grampa's relatives, Jed and Myra, such a return to the land is impossible after one has been exposed to "the finer things in life." "Linoleum becomes a habit," sighs Myra; "If I have ta give up eatin' them TV dinners I swear I think I'd die," adds Jed. The writers' highlighting of even at the time debased modern goods as symbols of modern living and the loss of a rural ethic offers a limited questioning of the "costs" of progress, and the storyline does reflect (albeit in an offhand manner) the exploitation of the land and people of Appalachia by extractive industries. In the end, however, this conflict between industrially driven consumerism and a pastoral ethic is finessed: Grampa agrees to sell the land to the box company and the factory owner agrees to relocate Great Grandma's house to the top of the knob where she can look out over the whole valley (although there is no mention of the fact that it will probably be choked with smoke!). The show thus both raised questions about the price of progress and resolved the issue in a way that ultimately upheld the primacy of mass consumerism and industrial development.

The underlying feeling is clear: changing your lifestyle is disruptive. Humans are creatures of habit, we form ourselves to our environment. Being shifted to another environment seems to produce feelings of nostalgia pretty much universally. If we go all the way back to even the Epic of Gilgamesh - the oldest written work - what we find in places are the character's nostalgia towards a nomadic lifestyle and a rejection of agrarian forms of living. Even today, there's no shortage of marketing towards nostalgia. The Real McCoys taps into those feelings, with the context of the inevitability of economics.

In the Andy Griffith show, the feeling was less sympathetic:

Usually rough-mannered and slightly unkempt older men, the few hill folk represented in the show's first two seasons served as unsophisticated and slightly antagonistic country cousins to Sheriff Andy Taylor (Griffith) and the Mayberry townsfolk, who were themselves perceived as bumpkins in the eyes of most urban outsiders on the show. Symbols of stubbornness, ignorance, and distrust of modern science and technology, the mountaineer characters tended to oppose any efforts to improve their lot in life. In a 1962 episode, for example, mountain farmer Rafe Hollister refuses to get a tetanus shot until convinced to do so by Andy, who sings the song he plans to perform at Rafe's funeral.

In "A Feud is a Feud" (1960), with a plot based loosely on both Romeo and Juliet and the love affair between Roseanna McCoy and Johnse Hatfield of the famed feuding families, two mountain farmers carry on a senseless conflict, the origins of which neither can recall, and forbid their children to marry one another. Although both men make a pretense of wanting to kill the other, they are afraid to risk their lives to accomplish the task. Here the older generation of mountaineers are presented as incompetently violent rubes, but their far-less-stereotyped children represent the capacity of such people to bridge the gap between an archaic and wrong-headed "mountain culture" and an idealized contemporary America.

On a particular few episodes about "mountain people" not fitting into urban areas:

The timing of these "mountaineer" episodes suggests they were not randomly selected storylines but were shaped by and reflected their historical context. ... the Darling and Bass episodes were also a response to the sudden reemergence of the Southern mountain region and people in the national consciousness and the conception of Appalachia as a distinct "problem region." These developments were in turn part of a general rediscovery of poverty in America in the early 1960s, a troubling realization that challenged the middle class's widespread faith in a classless society and an ever-improving quality of life..

Television comedies like The Andy Griffith Show and The Beverly Hillbillies that featured mountaineer characters reflected the national media's fascination with this "white other"-an isolated population outside mainstream American society. But they also served as a palliative for these disturbing images, providing a far more upbeat portrayal than did the news media of plain but comfortable folk who were upright, self-assured, and imbued with a strong cultural tradition. Such images lessened the sense of the deep failure of the American economic system and its inability to prevent seemingly intractable poverty. They also tacitly reinforced the widespread belief that the mountain people were trapped in a "culture of poverty," but redefined it from being a crippling cycle of degeneracy and dysfunction to a lifestyle choice of people who valued autonomy and family time over material advancement.

The Beverly Hillbillies was kind of a revolution in how they were presented. For one, it was popular among the audience it was designed to represent:

Such overall ratings do not necessarily indicate that a broad cross-section of Americans was viewing the show. A disproportionate number of the show's viewers lived in rural, small town or Southern locales...

Seemingly, it did this by sanitizing the representation of such people:

The show did move dramatically, although incompletely, away from the standard hillbilly stereotype. The first time the audience sees Jed Clampett, he rushes in the door of his bare-bones Ozark cabin and immediately washes his hands. Long flowing beards and outhouses never appeared on the show, nor did family feuds or shootouts with law enforcement agents, though Granny does occasionally raise her shotgun and rail against "rev'noors" and "Yankees".