It's a style of presentation that's very common in magazine articles.

The big boxes are there to give an idea of the most interesting, most important, or most attention-grabbing things in the article, for rapid skimming. (In magazines, I think they're there to grab your attention as you leaf through trying to find something worth reading. That doesn't apply in quite the same way to an article on the web.)

So the intended use is that (1) you skim quickly over the article, looking only at the stuff in the boxes, to decide whether it's worth reading, and/or (2) you read the article normally, basically ignoring the boxes.

It's called a pull quote, and yes, it's pretty standard for long magazine articles and their webby counterparts as an aid to skimming.

Usually, though, they're quotes from other people, not the author's own writing (maybe possibly if you have an editor, like in a magazine, it's not so bad). Doing it with your own quotes seems kind of pretentious.

And it's usually not placed directly below the actual line of text where it appears in the article.

It's part of magazines, practically any article over 2 or 3 pages will have them.

Correct but the article would have been a better read if the author didn't put the exact same words he used in the article in the quote and than place the quote behind the original text.

If you don't quote, it's not a pull quote, it's a heading or an aside. The example given is a bit of a miss, yes, because the same words appear quoted immediately; normally you'd pull something a little deeper in the text (like from the middle of a paragraph).

That's not quite right. A pull quote can also simply refer to an excerpt "pulled" out of the article text. Google it.

I did mention that the pull quote quotes (excerpts) the main article; that's what makes it different from a block quote. Though there may be ellipses, a summary is not a pull quote.

Actually, pull quotes almost always precede the quoted text in the article. They're entry points for readers to jump into the article, consequently it makes no sense to place them after the quoted passage. Moreover, they're secondarily used as a spacing device, namely, to expand an article so that it ends at the bottom of a hard-copy page, rather than being followed by empty white space. On web pages, of course, this is rarely necessary and so almost never done.