I imagine the layers to be mounted on wheels which are physically turned. Not all of the combinations yield a result, of course. Sometimes nothing happens. But there are usually multiple branches available, and the story will hint at which they are. The player can control both what happens in the story, and what memories are brought up. The game leaves it ambiguous whether memories that are not played out do not happen in the continuity of a particular play-through, or are avoided by the protagonists.
So the story may suggest that something terrible happened behind the garden shed back in 1957, and the player can switch to that combination, but the player's exact choice of the third layer, the characters, changes the story. Another less terrible memory may be alluded to in the same scene that suggests the shed, and the player may choose to visit that one instead.
As the plot progresses, options for layers are added and sometimes removed, so the player's choices also open up new characters and environments. Removal of options is more rare, but perhaps the plot moves on from that fateful day in '57, and that background becomes unavailable.
Finally, because the layers are physical things mounted on wheels, the characters themselves may sometimes interact with them, trying to move to a different scene. Here, the player can discover that they can block the characters' actions, stopping them from changing the topic, and forcing them to play out a particular scene.
All in all, the game mechanics and presentation are intended to be congruent with the story's topic: choosing which thoughts and memories to visit and make real, and which to avoid.