As you fight through each new area of the map, you will come across rooms that appear with stars and a level recommendations. These are the game's "Battle Stages" and where you will complete a lot of fighting.
Once you clear a Stage of enemies, you will be ranked based on certain criteria, time being the most obvious one. Basically you want to clear each of these stages as fast as possible (while also using things like special abilities, keeping your combo meter high, etc).
All you need to do is get an "S" rating (the highest rating) in each Battle Stage to unlock this trophy. I unlocked this one the first map of the game after the prelude with Gwendolyn. If you find you are having difficulty you can always play the story more and return to any map after you've loved up some. This is aalso a great way to grind for XP, Phozons, Items etc.
If you're curious how long you have in a Stage, check out the bottom right of your screen in a Battle Stage. You will see a timer that shows you how long you have til you are reduced in Rank.