The Purpose of the Battle Box is to Keep it For your Competitive team. Let's Say that your Party is For In-game and you like to battle Competitively but you don't want to switch From your in-game to your Competitive team? Don't worry Just Put your Competitive Pokemon in the Battle Box and you'll be Good2Go without switching from In-game to Competitive or you can use it For an XY Competitive team if you Battle people on Pokemon XY and then you can use your Party For an ORAS Competitive team so you don't have to Quit out of the Battle Menu.
Hope this helps :)
