When you start the game, you’ll need to pick Battles of Kyrat from the main menu. This will give you five choices:
Quick Play
Outpost
Demon Mask
Propaganda
Custom Match
The three game types you need to play are Outpost, Demon Mask, and Propaganda; you can ignore Quick Play and Custom Match. Simply pick each of those three modes in turn, and finish the game. You to not need to win, and it's okay if you join mid-game.
Once you’ve played all three modes, you will earn this trophy when the scoreboard appears for the third match.