Though I think the primary way that they could be used is a quick way to access saved data for particular characters, particularly in local co-op sessions. (nintendotoday.com)
A backstory is told via four sets of collectible audio logs hidden amongst the environments including recorded dialogue from characters about particular characters and certain situations. (thezombiechimp.com)
As the story progresses, one can develop particular characters, gaining access to more powerful combos that can be used in all the battles to come. (gamepressure.com)