Recently, I have spent some time reading, practicing, and watching the benefits of performing the snatch from just below the kneecap.  Previously I have always debated and wondered as to the usefulness of this position.  I always thought it was strange to snatch from there as it just seemed that in order to perform the movement correctly I would have to get the bar in the proper position at the top of the knee first.  So why not just snatch from the top of the knee?
Well as it turns out the reason this position is so useful and the reason we see it so often in international training videos is because it forces the lifter to get the bar in the proper position at the top of the knee FROM the bottom of the knee.  Basically, this position is HUGE for teaching the transition from the first pull into the violent, dynamic second pull.  Working the snatch from here forces the lifter to transition the bar to the hips properly AND gives room to teach a vertical track up the body.
The two biggest mistakes lifters makes ESPECIALLY  Crossfit athletes is (1) not performing the first pull and (2) swinging the bar out away from their body at the hips.  This position teaches correct bar path and movement.  Being below the knee forces the lifter to initiate the movement with the leg drive which encourages hamstring development and tension.  Secondly, this position forces the lifter to finish with a vertical second pull encouraging efficiency in their finish and third pull.
Try this position out as I have seen it make an impact in where I want the bar to be at and the tension I want to feel during the lift.  Here are three videos of some of my lifters doing them today.  There was some frustration by all of them by eventually they began to get the hang of it:
Natalie McClain Snatch Below The Knee
Lindsay Marshall Snatch Below The Knee
Emily Kelp Snatch Below The Knee
Also Coach Harvey Newton took one of my snatches and did an incredible job showing the proper positions for this movement in this video.  Check it out and be sure to click the tabs to see each position in freeze-frame.  He is an awesome coach and the one who sold me on this position as much as I hate doing it.