They "break up" to keep their relationship a secret.  Siblings can't actually marry - they're very aware of that - so their plan is to pretend to break up so that no-one can be critical of them.
It's a funny relationship with the beginning of the anime, where Kirino is keeping a secret, but is later revealed to everyone.  Here, wee have something that's already known by everyone else (before even the couple in question) but later becomes a secret.