I come from an AL city also, but this is my understanding of how & when it would be done:
You would perform a double switch when your team is on defense, you want to make a pitching change but your pitcher is due to bat soon.
So lets say you still have your starting lineup on the field in the bottom of the 7th, you want to pull the starting pitcher for a relief pitcher to face the opponent's star hitter. Your left fielder who was the #8 hitter made the last out of the last inning so your pitcher is due to lead-off the next inning.
What you do is make 2 substitutions at once, and switch the player's positions.
1. You put in a substitute outfielder for your pitcher - in the #9 spot of the batting order.
2. You make another substitution where you bring in a relief pitcher in place of the left fielder - he takes the #8 spot in the batting order.
But you have the switch positions so the relief pitcher (who is now in the #8 spot in the batting order) takes the mound and the substitute outfielder takes his place in left field.
Now you have the correct pitcher on the mound but when your team comes to the plate the following inning its the subsitute outfielder leading off in the #9 position and you don't bring a pitcher to the plate again until you get all the way around to the #8 spot in the order again.