Fiction-first is a bit of jargon to describe the process of playing a roleplaying game, as opposed to other sorts of games you might be used to. In a standard board game, for example, when you take your turn, you choose a move from one of the mechanics of the game, and then use that game system to resolve what happens. You might say, “I’m going to pay two stone to build a second fort on my home tile.” We could call this process “mechanics-first.” What you do on your turn is pick a mechanic to engage, then resolve that mechanic. Your choices are constrained by the mechanics of the game. You might color it in with some fictional trappings, like, “The brave citizens of Baronia heed the call to war and build a stout fort!” but the fiction is secondary; it’s flavor added on. In other words, the fiction is brought in after the mechanics, to describe what happened.
In a roleplaying game, it’s different. When it’s your turn, you say what your character does within the ongoing fictional narrative. You don’t pick a mechanic first, you say something about the fiction first.