These are a Meta-Currency that allow players to interfere directly in the game mechanics and the direction of the story. Why in god's name would you let them do that? Because this game is about the characters' story, not about preserving the sanctity of your plot, or making sure the rules are enforced. Remember, there is no cheating. There is only making things more epic.
FATE Points (FP) are granted to players for willingly suffering defeat, for playing their characters, not gaming the rules, and for making the choice that enhances the story, NOT the smart, selfish choice.
Every character starts each session with at least one FP, unless the last session ended in the middle of a scene, or it makes sense that there has been no chance for them to catch their breath.
FP can be used in a number of ways: The player can spend one to completely re-roll their FATE Dice, using the new result. They can add +2 to the final dice roll, after any re-rolls have been done. Some Mortal Stunts and Supernatural Powers require a FP to activate. They can make a Declaration. Most of these are pretty self explanatory.
Making a Declaration is much more interesting. The player may offer the GM one of their FP to Declare that something is True. If the GM accepts it, the Declaration becomes True, with no chance of failure or the kind of nasty dealing that genies granting wishes are known to indulge in. The declaration becomes true in the manner that the player intended it, not the way the GM can think of twisting it.
This could be as simple as Declaring that the character had spotted a fire extinguisher in the corner of the room, and blam, there is one. It could be much more complicated however. The player could declare that they had previously met with a member of the foreign peace delegation, the clerk in charge of records, and slipped him money to alter the list of allowed attendees at a cease-fire summit to include the character and his group. As a result, the characters were sent real invitations, and now the massive guards at the door will accept the invitations and welcome them inside the impregnable fortress without any fuss. The GM considers, decides that it is something that the character might have thought of, and has the connections to pull off, and accepts the FP. Blam, reality is something different than it was.
The rule to this is that Declarations should improve the story for everyone. The player that Declares that the BBEG suffered a sudden heart attack and died, leaving his organization in chaos and destroying the plotline is going to receive hard looks from the GM and the other players, and the FP will not be accepted.
Just as FP are considered a narrative currency used to trade concepts for power, an economy of their trade will develop as the characters start to get into the story and make changes in what happens. Like any economy (let's just stipulate that I'm right and not have the pure capitalism argument) it needs careful management by players as well as the GM. FATE is flexible enough, and people are creative enough, to imagine ways to request FP in almost any situation. On the other hand, it's not just the GMs responsibility to recognize when someone is deserving of a FP for their dramatic play or creative problem solving. Players need to be able to recognize the times when they deserve recognition and ask for it. They need to point out situations when other players are deserving. The FP economy needs to enhance the drama and excitement for everyone.
It's tough to quantify guidelines, let alone rules for the giving and taking of FATE Points, but players need to feel like they have real, immediate control over the narrative, while at the same time they shouldn't be able to throw FP at a problem until it goes away. When players have a surplus, they are inviting the GM to up the scale of the opposition, and accept player actions that are an order of magnitude more grandiose. "Hit that guy with a bat? Why not pick up a car and use it like a bat?" Making the opposition bigger / more powerful requires more FP to bring the characters up to that level. It's a good way to make the game feel more exciting, and for the characters to feel like they were able to pull out all the stops when they really needed to. Just be careful not to force feed them FP. Nothing kills the fun like power they didn't have to earn.
At the same time, you don't want to drain a lot of FP right before the climax of the story, unless your story is about endurance under punishing conditions etc. In my experience, players want the conclusion to be epic in scale compared with the rest of the story, not just have their characters be able to survive long enough to win. Your mileage may vary.
Ttory_Seller had this to say:
As a GM, I will say that I greatly appreciate players being willing, and cognizant of their Aspects to the point where they will self-compel, or remind me when an Aspect of theirs is relevant. For instance, in the game I'm currently running I have 5 PC's, a hostile environment (weather and monster), as well as location Aspects to juggle. Oh yeah, and the story to narrate and propel forward. I'm sure that I don't always remember Aspects to compel to make things more interesting, though I do try. Having players who are equally invested in the story and their characters to the point where the will (hopefully temporarily) make things more difficult for their characters makes the story more exciting, interactive, and hopefully move forward.