Completing this achievement will also unlock Wrecking Ball and Tourist.
A level transition is when you knock your opponent into another part of the stage, triggering a cutscene and causing additional damage. To perform a level transition, you first need to push your opponent into the corner of the stage. It can be the left or the right depending on which stage you're on. Once you have your opponent in the corner, press backwards+A . You will then knock them into the next part of the stage if there is a level transition available in that corner.
Most of the stages have 2 transitions. Metropolis has 3. While 'Atlantis' and 'Ferris Aircraft' have none. Below is a list of the stages and which corner of the stages the level transitions are on.
Batcave: Right -> Left
Gotham City: Right -> Left
Themyscira: Left -> Right
Arkham Asylum: Right -> Right
Joker's Asylum: Right -> Right
Insurgency: Left -> Left
Watchtower: Left -> Right
Stryker's Island: Left -> Right
Hall of Justice: Right -> Right
Fortress of Solitude: Right -> Left
Metropolis: Right -> Right -> Left
Wayne Manor: Right -> Left
Wayne Manor Night: Right -> Left
This can be done through normal gameplay or much more easily in Practice Mode.