What’s is this device and how should it be used? It seems to be to provide emphasis of some sort, but so often it seems completely out of place. Sometimes it seems to be to used as a way to insert something into a quote that wasn’t said originally, but definitely not always.
And before someone tells me to try the search engine…just think about that for a second 
Not entirely sure if this is correct, but…
I have seen it used in two cases:
a. the original wasn’t entirely clear, and the quoter wanted to clarify the ambiguous (sp?) phrases.
b. the original was missing a few words/spelling errors.
Use 1 that I’ve seen:
<source text>Andrew Arnold was scheduled to speak, and waited through his introduction for approximately twenty minutes. But then he decided to take a walk, and disappointed many fans. He left in his private jet shortly thereafter.
<my quoting of>I was at the rally, waiting for Andrew to speak, but the first speaker would not shut up. Finally, I gave up and went home, and according to the newspaper, “[Andrew] decided to take a walk” and just frikken took off! Jerk.
Use 2 that I’ve seen:
<source text>Andrew Arnold was scheduled to speak, and waited through his introduction for approximately twenty minutes. But then he decided to take a walk, and disappointed many fans. He left in his private jet shortly thereafter.
<my quoting of>He was hanging out back stage, from what I heard, and the website confirmed it: hanging out and “[T]hen decided to take a walk, disappointing many fans.”
User 3 that I’ve seen:
<source text>“It was like totally cool the way the rains could like control where the horse was going!”
<my quoting of>She went on and on about her first time out, writing in her journal: “It was like totally cool the way the rains [sic] could like control where the horse was going!”
<my quoting of #2>She went on and on about her first time out, writing in her journal: “It was like totally cool the way the [reins] could like control where the horse was going!”
Fixed #2 more better (but not much) 
Well typically as long as you are consistent throughout a work you can use virtually any symbol to mean virtually anything.
Somethings have been codified by certain conventions or groups. For example the MLA, APA styles specifically explain how parentheses ( ), and brackets are supposed to be used.
The most common usage for brackets appears to be making a correction in a direct quote. Like if Thomas Jefferson gave a speach about the Cherokee Indians in which he said, “The Cherokee are a brave and mighty tribe, with a proud 400 [200] year history.”
The above correction is saying that Jefferson was wrong, and at that point the Cherokee only had a 200 year history. The correction can be grammatical or factual. Typically grammatical or vocab errors in quoted text are just corrected by using “[sic]” so it tells the reader the author of the work recognizes there is a grammatical error in the quoted passage and that said error is part of the original quote.
I’ve never seen brackets or parentheses used for emphasis. I’ve seen parentheses used heavily to make asides, and brackets much less so.
Another use I’ve heard suggested as being correct is to sort of take the place of an ellipsis. For example, “I knew doctors. They often worked overtime, but didn’t get paid. Despite this, they were some of the most charming people I’ve ever met” becomes “I knew doctors [who] were some of the most charming people I’ve ever met.”
It’s better for an essay where the reader has the ability to look up the quotation, based on the author’s citation.
I think what you are referring to are clarifying words insterted into a quote. For example if I said:
That couldbe shortened up to:
Just using “It” would not give enough context.
“They’re used to paraphrase or interpolate, when the original was misleading or incomplete or just rambling on and on.”
“[Brackets are] used to paraphrase or interpolate [part of a quotation], when the original was misleading or incomplete or [wordy].”
I agree with Dan and Tenar.
I’m not sure one can do what Speaker for the Dead has done in his(?) presumably invented example. The basic rule is “square brackets to indicate insertions into the original text; ellipses to indicate omissions from the quoted text”. (“Who” in the cited case would count as an insertion, but since an omission is also involved, perhaps it should be preceded by three dots. Better to rewrite it somehow, perhaps using reported speech or dropping the “I know”.)
However, this brings me neatly on to a consideration of what might be called the journalistic use of square brackets, to render an otherwise ungrammatical, or infelicitous, text grammatical. It is quite often done when the utterances in a spoken interview are rendered in print. Thus, if a football manager says “played very well” in an interview, it might be rendered as “[Keane] played very well” by a conscientious journo.
Yes, I tend to go for … or [ … ] when I’m chopping out big whomping bits of text.
I can look it up later on, but I belive the current MLA format actually requires the use of brackets around ellipses in quotes when the writer omits words in an original quote.
E.g. source: “Once, in 1960, for 20 minutes.”
quote as used by writer: “Once [. . .] for 20 minutes.”