"Lessen, poisoned gulls, ditcher wander hair annulled furry tell a boarder Slipping Booty?"

For a little more context, the preamble to that story in the article you linked says: [Edit: I took this long block out since it was included in cornbread ninja's answer as well.]

The first line:

Lessen, poisoned gulls, ditcher wander hair annulled furry tell a boarder Slipping Booty? Hoecake? Wail, heresy starry.

can be "translated" (to the best of my ability) as:

Listen, boys and girls, did you want to hear an old fairy tale about a Sleeping Beauty? OK? Well, here's the story.

The adjective used in the preamble is frammis (a nonsense word you use when you don't know the real word for what you are describing), but you could describe the technique used to create the story as a stringing together of deliberate mondegreens, or a homophonic transformation. It is not generally used for anything other than amusement.

There is a popular party game here in the US (and elsewhere, I imagine) called Mad Gabs, which utilizes similar constructions.

The link you provided goes to page 4 of the entire article. The following paragraph precedes what you have quoted and I believe is quite helpful:

Slipping Booty

Five years ago this week the first issue of SPORTS ILLUSTRATED was published and, in the line of parlor games, we included a highly improbable fiction called Ladle Rat Rotten Hut by Howard Chace, a language teacher at Miami University, Ohio. With an insidious knack for distorting the spoken word when putting it on the printed page, Chace blithely started his story, "Wants pawn term" (for once upon a time), and let the reader struggle on from there. Read aloud, and preferably in company, Professor Chace's frammis constructions made a sense of their own. So much so, indeed, that a collection of them—titled The Anguish Languish—was published in book form by Prentice-Hall in 1956.

Now, as an aunt a verse ray (excuse us), hello, Professor Chace has contributed another fairy tale as it might be told by a psychotic speller with a speech impediment. Here is his not-so-Grimm rendering of Slipping Booty. With the helpful hint that "Lessen, poisoned gulls," should be translated "Listen, boys and girls," we turn you loose to see how fast you can figure out what really happened to the handsome prince:

So they are frammis constructions.