Had this discussion a couple of years ago and did some research... It apparently goes back to old rules (1800's) where an out was only recorded if the ball was caught.
The dropped third strike is more properly called "an uncaught third strike". If you look at it that way it makes more sense.
The only ways to get a batter out are to catch a ball, tag him with the ball, or for the ball to beat him to a base that he must run to. An uncaught 3rd strike doesn't fill any of those qualifications.
The rest of the rule (1st base unoccupied part) is to prevent a double play by a catcher intentionally dropping the third strike - along the same lines as an infield fly.