Here is the relevant rules text, from p 6 of In A Wicked Age (I've elided some stuff that deals with particular mechanics and examples):
Naming your characters’ best interests
GM, you start. Start with one of your strongest NPCs, and name two best interests for her. Each should represent a direct attack on at least one player’s character.
From there, you and all the players take turns, until every player’s character has at least two best interests named and every NPC has at least one (and many have two). Just jump around the circle, with you as GM taking a turn whenever you like.
For players
. . . You can cast your character’s best interests against your fellow players’ characters, or against the GM’s characters, freely, without any distinction between the two. . . .
Your character’s best interests don’t even have to be mutually compatible. Since their value is in the conflicts they create for her, you don’t have to worry about whether they’re even achievable, let alone achievable together. . . .
For GMs
. . . You can – and should! – ruthlessly threaten the players’ characters, where they’re weakest.
It’s fine to occasionally cast one of your NPCs against another, but it doesn’t do anything for the game. If you find that you’re naming NPCs in your NPCs’ best interests, STOP. Go back and do over.
Look for hooks and curves. If on paper a player’s character and an NPC are allies, think flexibly and find a best interest that threatens or undercuts their alliance. . . .
You’ll take the lead by naming best interests that are direct threats to the players’ characters. If you’ve led strong, look for the players to react to your attacks, then pass new attacks on to each other.
Jump in with your NPCs’ best interests opportunistically, whenever you can add threats, complications, or the potential for violence, or whenever two players’ characters are starting to look too cozy together. If you look for the angles, the leverage you can use to wedge them apart, you’ll find it.
"Best interests" create a network of conflicts, and in this way they provide the basis for scenes/situations. From pp 10-11:
GM, it’ll mostly fall to you to open and organize scenes. . . .
Start the chapter by zooming in. . . . Also, start the chapter with at least one player’s character, always, and start with a meeting. Choose characters with something to say to one another – characters with a chain or knot of best interests between them. . . .
To rush up to a conflict:
Choose two characters who want to do harm to one another. Arrange circumstances so that one of the characters has a sudden, momentary, immediate advantage – act now and seize it, or hesitate and lose; and furthermore that if she
should leave her enemy capable, the advantage will dramatically reverse. . . .
To circle a conflict:
Choose two characters who want to do harm to one another. Arrange circumstances so that they have to interact, but so that neither of them have any upper hand, and in fact so that if one should attack the other, she will do so at a significant disadvantage. . . .
To draw a conflict out:
Choose two characters who want to do no harm to one another at all, but whose interests don’t mesh well or overlap. Arrange circumstances so that one has the opportunity to pursue her interests, but only by threatening the interests of the other. Also arrange it so that the other will see her do it, or have evidence that she’s done it, or have some reason to blame her for doing it – so that the offense is unignorable.
So "best interests" are a particular way of using motivations, to structure and progress scenes/situations in play. I think the In A Wicked Age rulebook sets this out really clearly, but it's an idea that can be generalised.
If the GM keeps the NPCs' best interests secret, or makes them hard for the players to know, then the players are likely to find it hard to anticipate what scenes will occur, and to understand what is at stake in the scenes the GM frames. If, as in In A Wicked Age, the NPCs' best interests are known by the players, then the players have a sense of what is coming, and what is at stake.