You get Fate Points from adverse invokes at the end of the scene.
Here, under invoking aspects:
If the aspect you invoke is on someone else’s character sheet, including situation aspects attached to them, and the invoke is to their disadvantage, you give them the fate point you spent. (Invoking a third party’s aspect is treated just like invoking an unattached situation aspect.) They don’t actually get to use it until after the end of the scene, though.
This is to prevent the same point bouncing back and forth until someone runs out of aspects to hit with it.
You get Fate Points from conceding conflicts at the end of the conflict, which should be the end of the scene.
Here, under conceding the conflict: (emphasis original)
Concession gives the other person what they wanted from you, or in the case of more than two combatants, removes you as a concern for the opposing side. You’re out of the conflict, period.
But it’s not all bad. First of all, you get a fate point for choosing to concede. On top of that, if you’ve sustained any consequences in this conflict, you get an additional fate point for each consequence. These fate points may be used once this conflict is over.
In general, a conflict will be notable and impactful enough to end most scenes it occurs in, but there might be circumstances where you want to quickly wrap something up in the aftermath.
You get Fate Points from compels immediately, but compels will often start or end a scene.
Specifically, there are two types of compels: compels that create events, or compels that force decisions. Both of these types of compels can be used to create opposition where there was none before, or break a scene decisively in the opposition's favor, which will start or end a scene.
During a scene, both types of compels can also be accepted to escalate something already in progress.
A footnote: the GM's Fate Point budget
There is not a finite number of Fate Points. Here, from the GM and Fate Points:
When you award players fate points for compels or concession, they come out of an unlimited pool you have for doing so—you don’t have to worry about running out of fate points to award, and you always get to compel for free.
The NPCs under your control are not so lucky. They have a limited pool of fate points you get to use on their behalf. Whenever a scene starts, you get one fate point for every PC in that scene. You can use these points on behalf of any NPC you want, but you can get more in that scene if they take a compel, like PCs do.
"Keep the Fate Points flying" more likely refers to the use of the unlimited pool to keep the plot moving with event- and decision-based compels.