For this achievement, you have to beat the story mode for three different play sets. You don't have to 100% them. Instead, just keep playing missions until you progress the story to its end. This will unlock as you watch the third Play Set's ending. Since the game comes with three, just beat those to unlock the achievement.
Just an FYI, story missions are given to you by the main characters of the movie, not the random people walking around. For example, a random tentacled monster's mission isn't story while accepting one from Terry & Terri is considered a story mission.
You'll unlock the following along the way as well: