I know it is specific to the National League home games but what is the purpose/strategy and the rules of a double switch in Baseball?
The double switch is used, primarily (but not exclusively) in the National League, when replacing a player for defensive or pitching purposes who is a poor(er) batter, to place them further from the current batter than they normally would be.
For example, in the top of the eighth inning, two on, two out, you replace your pitcher, batting ninth. In the bottom of the eighth inning, the seventh batter will be coming to the plate to lead off the inning. As such, your new pitcher would now be batting third; that's not ideal, since he's not a very good batter, and so you'd need to either give up a free out, or pinch-hit for him.
Assuming this is a pitcher you want to pitch the top of the ninth - say, this is your closer, and you're up by a single run - you would rather move him up in the batting order, say to 6th. That way he's unlikely to come to the plate in the eighth, at least unless you score a lot of runs (in which case you're probably fine with either result).
As such, you take out your left fielder, who is batting sixth, and replace the pitcher in that spot in the lineup, and a new (perhaps defensive-minded) left fielder in the 9th spot in the lineup. Technically what you're doing is substituting the new pitcher for the old LF, and the new LF for the old pitcher; but it gets your batting lineup correct. Now you have an at least somewhat competent bat up third the next inning, and your pitcher won't bat.
This is covered in part in rule 5.10:
If a double-switch is being made, the manager or coach shall first notify the plate umpire. The umpire-in-chief must be informed of the multiple substitutions and interchanged batting order before the manager calls for a new pitcher (regardless of whether the manager or coach announces the double-switch before crossing the foul line). Signaling or motioning to the bullpen is to be considered an official substitution for the new pitcher. It is not permissible for the manager to go to the mound, call for a new pitcher, and then inform the umpire of multiple substitutions with the intention of interchanging the batting order.