How was popular sovereignty supposed to work?

In addition to the good information in the above post,
"popular sovereignty" was part of the compromise hammered out in 1850, largely by
Senator Henry Clay, which allowed Texas in as a slave state, and Maine as a free state. 
This maintained the balance in Congress between free and slave
states.

The westward movement and Manifest Destiny had
thrown a monkey wrench in the balance, and the old Mason-Dixon dividing line between new
free and slave states wouldn't work anymore after the Mexican-American War.  Almost all
of the new states from the Mexican Cession would end up being slave states that
way.

So popular sovereignty imagined that once a territory
reached 50,000 residents and qualified for statehood, an election could be held where
the majority rule of the new state could decide its slavery status.  This seems
democratic, and respectful of the culture and opinions of the territories that were to
enter.

Its authors did not foresee the large influx of
radicals from both pro and anti-slave forces that would bring so much violence to
Kansas.