There are a couple things to consider here: first, players in Fate are encouraged to narrate as much as possible when it comes to the story as it involves their characters. Second, the GM has the last say at the table, but that might not mean what you think it means. Now let's go into more detail.
Fate wants your players involved with the story, both in creation and telling. When the players work with each other and with the GM to describe the results of their actions, it enhances the experience for everyone involved. But it is that: the players are describing the results of their actions. The dice have already fallen, success or failure has already been determined, they're just describing what that looks like. When they want to declare details beyond what have already been described, it's time to break out the fate points. There are a few ways to accomplish this as presented by the mechanics:
- Create an Advantage. This is one of the foremost ways to change facts about the game, especially as it involves conflict with other characters. By creating aspects and boosts on characters and the environment, your players are quite literally helping to define reality. Related to this is:
- Declaring a Story Detail (Fate Core p13). This is essentially trading a fate point for the ability to declare something true in the story, as it relates to the character's aspects. A cat burglar spending a fate point to declare that of course they brought along glass cutters is perfectly reasonable and justifiable. The same cat burglar declaring that of course they have a machine gun stowed away in their pack? Maybe less so.
- Compelling and Invoking Aspects. There is plenty of story to be told as it relates to the facts that have already been established. It sounds like your players (or at least the one in question) already understand this, but it bears repeating: if characters are creating complications (including for themselves) or leveraging established facts to further the game, fate points should be changing hands.
GM's Last Say
Since your players have several mechanical outlets available to accomplish their goals, it's your job to keep them in check, so let's talk about what that means.
First, let's talk about what a fate point is. A fate point is, essentially, currency for narrative control. Your players spend them to describe narrative (as discussed above), earn them for giving up that control, and it's important to note that you as the GM have an endless supply of them. That means that if you are inclined to be heavy-handed, you will always be able to "out-spend" your players. Part of the art of GMing a Fate game is knowing how to move the story forward by introducing complications (compelling players) while encouraging the healthy flow of fate points in the other direction.
As the GM, you have the last say, but that right should be exercised as much as possible within the fate point economy and rarely, but rarely, by fiat, unless there is an obvious problem with balance or power levels or what have you (and even that bears discussion with the table to see how the other players feel about it).
Part of your question was about whether you should write down what you've prepared for the players. To this I say absolutely you should. The more your players know about the NPCs and environments they face, the more they can help you in telling the story that you all want told. This doesn't mean that you have to tell them everything (surprises are still a part of telling a story), or that their characters know everything: part of the purpose of the Create an Advantage action is to uncover aspects that already exist.
The other part of your question involved how much prep work you should be doing. That's more of an art than a science, especially as it relates to Fate. Some GMs have nothing more than a few index cards for characters with aspects, skills and stunts, which is a completely valid approach that requires more thinking on your feet and gives your players more control of the story. It's probably a decent rule of thumb that the more prep work you've done, the more you'll be compelling players to keep the story flowing the way you've planned it out, but even in this case, you should be as flexible as possible in allowing players to influence the story the way they want to go. That's what fate points are for, and the game provides lots of mechanical ways (such as conceding in a conflict instead of being taken out, for villains and such) for your plot ideas to "live to fight another day," even if they require a little cosmetic surgery to fit with how the story has unfolded.