"Advanced" stats in hockey are not all that advanced from a stats point of reference. It is all about counting things, and then dividing them out. For example, if a player puts 10 shots at the opposing net while the other teams put 8 shots on net while he is on the ice the Corsi works out to 10(his shots)/18(total shots for both side while he is on the ice)=55.56%.
Sometimes you don't count blocked shots, and sometimes you standardize by figuring out what things work out to in a 60 minute game, but it doesn't go too far beyond that at this stage.
The idea is that the other team can't score when you have the puck, and when you have the puck more you end up directing more shots at the other team's net. Over the course of time the teams who have the puck more tend to win more often that teams that don't have the puck as much as their opposition.
There is a lot of common sense behind it, and it really revolves around just counting stuff that you see on the ice. "Advance stats" as a term really only makes sense in comparison to the older stats, which focus on goals, assists, +/-, save %, and things like that. I think sometimes people see "advanced" and "stats" together and assume that it is something only brainiacs with computers and calculators can figure out, or they think the people who use stats focus on numbers instead of watching the games.