To get this trophy that's called "World's Greatest Detective", you must discover who is behind the spirit of Armadeus Arkham.
To do this, you will need to find all the Chronicles Of Arkham that are scattered all over the island and scan them all.
When you do this, the "spirit" implores you to search and find him.
The easy answer to this is back in the Penitentiary. It's back at the Security Control Room, in the little cell where Quincy Sharp hands you the card needed for the cryptographic sequencer that Batman uses for the first time in the story.
Climb up those stairs until you reach the end of the path and look down on the ground. You will see messages created on the floor, the center of it repeating "Batman" over and over again. Scan that final chronicle one more time and the answer to the mystery will be revealed.
Have fun and good luck! Here is a video showing you where you need to go and how to find the final Chronicle Of Arkham.