It's a progressive death spiral. Mass market fiction magazines don't pay enough to justify the time spent writing a story -- because their circulation and readership is declining. So short fiction has become the domain of people who don't write for money -- like poetry: once a respectable paying niche in newspapers, and now little more than a hobbyist field curated by a handful of scholars. And then you get the curious aridity King notes, an airlessness in the room, as the stories being published chase an audience of readers motivated by something other than entertainment.
If you want to read a great collection of short stories, go pick up "Everything Ravaged, Everything Burned" by Wells Tower. It's a fine debut.
Twain's a master of a lot of things in writing, obviously, but one of my favorite things he does well is the character of "pompous, erudite jerk who doesn't realize he's actually a moron". (Stephen Colbert is a good modern example of such a character.)
The Man-Kzin Wars series is an example of this run amuck: dozens of authors publish stories set in Niven's Known Space universe through this collection. They're an absolute joy to read and a great way to be introduced to many new authors.
http://www.amazon.com/Best-American-Short-Stories-2005/dp/06...
http://www.amazon.com/Maps-Legends-Reading-Writing-Borderlan...
His argument is, basically, that Modernism (here meaning the Joycean/Chekhovian "epiphany" story) has damaged the short story, or at least that it has played itself out, and that the old-timey plotted, adventure story looks fresh again.
But really, as others have mentioned, commercial short story markets have been in decline for 60+ years. Science fiction magazines are one of the last holdouts of the pulp era, and they're not all that healthy (commercially). This is making (has made?) general short fiction more like poetry. That ain't all bad. One upside may be that as commercial short fiction declines, writers and readers may be less inclined to think plotted short fiction == bad fiction.
Anyway, there is still a lot of quality contemporary-ish short fiction out there. If you like short fiction, you will certainly like some of it: Denis Johnson, Lorrie Moore, Alice Munro, Matthew Klam, Arthur Bradford, George Saunders, T.C. Boyle, and on and on and on.
My personal belief is that the short story, and maybe even the poem, is on the verge of a comeback based on the rise of the eBook in Kindle/iPhone format. I think soon we'll see business models whereby its reasonable for an author/poet to sell an individual short story or poem. And that people will pay to read them on their phone or eReader.
OTOH I'm biased since short form is the sweet spot for my textflows technology!