Battle Rating is a player's score that determines their Online Battle ranking, which determines the rewards they receive at the end of the season.
Battle Rating is received upon winning any Online Battle match, scaled to your opponents' stats. Completing a whole battle (except quick battles) will give a bonus amount of Battle Rating, with Ultimate Battles giving by far the most. Winning a defense match would award twice the normal Battle Rating. Additional Battle Rating could be awarded upon completing Bonus Missions, which is fixed and do not scale with your team's strength.