Why do you wear your coat like that?! For what purpose do you have one arm out of the sleeve and hanging there like that?! Was there ever a stated reason for this? Does he explain it? I've been confused for far too long...
The stance is a classic, badass Samurai pose. Just to add some random trivia to that point, I believe Toshiro Mifune was the one who trademarked the "arm in the jacket" thing as the quintessential badass samurai pose for years to come.
When I looked up, I got this.
Auron tucks his left arm into the front of his haori, making it look as though his arm is in a sling. He removes his left arm from his haori when fighting, but his right arm remains in its sleeve. This aspect (along with the jug on his side) is a tradition of sorts among rōnin, and is likely meant to be an homage to this traditional figure: a rōnin is a samurai who refused to commit hara kiri (ritual suicide) after his lord's death, maybe a reference to Braska's death.
Or in Itachi case, the death of his family/clan.
Bingo.
That's the reason. I'm surprised this isn't common knowledge by now, seeing as how it's so badass.
EDIT: For the record, I've heard that this arm-through-the-jacket thing is meant to look like the arm has been cut off/amputated somehow. Rounin supposedly do it to symbolize loss (i.e., the death of a master, or in Itachi's case, his clan), as stated above, like "an important piece of me is gone now" and such.