The blue stuff in the airplane toilet is a chemical known as "blue Ice" which helps break down the waste and helps with the odor in the toilet.
The airplane toilet is a vacuum toilet and uses an active vacuum instead of a passive siphon to flush.
When you flush the airplane toilet it opens a valve in the sewer line, and the vacuum in the line sucks the contents out of the bowl and into a tank.
The poop and waste does not get dumped out of the plane in the air because that's not allowed although if you do see some poop or something coming from an airplane it's likely just a leak from thew sewage system that does happen from time to time.