Stage striking refers to CPs. Each player strikes a CP, which means that the person selecting the stage cannont select a stage that has been struck. For neutrals, you have what is called a "random reset." Basically, both players can agree upon a neutral stage that they want to strike if it comes up during the first match. If the stage comes up, they redo the random selection for the first match.
Also, each player can use their strike on stages that is part of the neutral list.
This is the selection system that has been used during the most recent tourneys. If it is different, please direct me to a link with the specifics in it.