When you destroy the DUP headquarters in any given zone, the audio log mission for that zone will appear on you map and become available (One audio log mission per zone).
These missions are very simple, all you have to do is press up on your D-Pad and use the signal tracker on your phone to close in on the signal, once you get close to the audio log Delsin will collect it automatically. Once all of these missions are complete, the trophy will be yours.