Why do Jinchuuriki die when their biju are extracted? When tail beasts (bijū) are extracted from their Jinchuuriki, the Jinchuuriki dies. 

All seven Jinchuuriki died after the Akatsuki extracted the bijū from them. Gaara was resurrected by Chiyo using the "One's own life reincarnation" jutsu. 

So why do Jinchuuriki die when bijū are extracted?
What are the reasons behind the death of Jinchuuriki after the extraction of bijū?
The reason are not been clear, probably it never will be, now that Naruto is scheduled to end this year.

The Sage of Six Paths told himself that no matter what, if a Tailed Beast is extracted from someone, they die.

I believe that's canon material and a fact in Narutoverse. I guess we just have to make our peace with it.

On a personal note, in my opinion, when someone becomes a  Jinchuuriki, the huge chakra of the Tailed Beast enters into the ninja's body and gets mixed up with that of the ninja's.
Imagine the body of ninja is water, and the ninja's chakra is a pinch of sugar dissolved in the water. The process of turning into a Jinchuuriki would be like adding salt (= Tailed Beast's chakra) to the mixture. Now to extract the Tailed Beast's chakra, it's impossible to do so without drying up the ninja first, i.e. Ninja has to die.