I've heard a few different reasons people assign to the phenomenon, none of which, to be perfectly honest, have ever completely satisfied me as being the only correct answer.

The two most common explanations are obsessed fans and straight up greed- people looking to make some easy money by capitalizing on Elvis' image. While there is certainly some truth to both of these, as I stated earlier, I don't think either one of them tells the whole story.

Elvis was obviously an immensely popular figure, and his more devoted fans earned themselves quite a reputation for their obsession. (If you're interested in Elvis fans, Dave Barry wrote what he referred to as a "semi-serious" piece on Elvis fans. It was published in Dave Barry is not making this up, and it's quite an interesting read, he doesn't play into the stereotype of "fat, badly dressed Elvis fans" and manages to interview some people who camped outside of Graceland regularly while Elvis was alive.

The salient point is that even today Elvis fans are extremely devoted to his legacy:

The fans were there last year, they'll be there this year. This doesn't happen for the Beatles, it doesn't happen for Sinatra, it doesn't happen for Franklin D. Roosevelt. It doesn't happen for anybody I can think of that is not the focal point of a major religion. (His italics)

The problem with chalking Elvis impersonation entirely up to obsessive fans is that to do so, one has to assume that Elvis fans are more devoted to Elvis than fans of literally any other entertainment, which is just not true.
To quote Starlust, a collection of letters sent by fans to a variety of pop-stars:

it is not any particular fan who is extreme, so much as the condition of fanhood itself.

Any reasonably popular Musician, Movie or TV Series is going to have it's die-hard obsessed fans. If merely having obsessed fans led to imitation being widely celebrated, we would all be seeing John Lennon impersonators and Off Broadway adaptations of Season 3, Episode 6 of The Simpsons. (Yes, people do imitate things they are fans of, but imitation for fun and having an entire industry of people imitating professionally are very different things).

I've read a few papers (most for a sociology project I had to do in college) that flat out dismiss Elvis impersonators as hack singers looking to make an easy dollar.

Yes, people get paid to impersonate Elvis, but there are easier ways to make money than learning most of an artist's catalog, buying a jumpsuit and learning to sing and dance. Plus, for someone to get paid doing something, there needs to be someone willing to pay them to do it, unless it's a government job. (Zing!)

I would argue that one underlying reason behind Elvis impersonation's endurance is that in the US, 1950s nostalgia is a big money spender, and Elvis was arguably the most recognizable music star if we look back at that decade. So Elvis became one of the faces of the "good old days." His appearance remains recognizable, and would be without imitators donning a wig. For someone to be impersonated, the audience needs to be familiar with them to start with. (Dave Barry wrote a column on things from the 1950s becoming cool again as well, but there's not much academic value to it, and regardless, rumor has it that every time you cite a humor columnist on /r/askhistorians a moderator gets a migraine).

Another factor in play is that Elvis was somewhat of a joke in the press towards the end of his life. Overweight, drug-addled and in failing health, he was mocked mercilessly by the tabloids of the day, which helped keep a caricature of him in the forefront of the public mind, even as he went from being the biggest chartbusting artist out there to enjoying a more moderate level of success. (It says something about how big he was that it comparatively looked bad when he "only" released an average one top-5 selling album per year the last 4 years of his life). The public was already used to people pretending to parody Elvis, and some impersonators doubtless got parts of their act from these caricatures.

Ultimately, I think nostalgia and continued fandom are probably the two biggest factors- which is interesting, because for the time being, they are almost cooperative- 50s nostalgia helps create new Elvis fans, who buy Elvis memorabilia and/or go see an impersonator, which fuels the creation of 50s nostalgia merchandise, which helps create new Elvis fans...