Tell me — what's this: 
 
 
 
It's a circle, right? Well, no, it isn't: It's a crescent of varying shades of gray. But your brain and eyes don't work like that. They were designed to see things, not marks. Artists spend years learning to decompose what they see into constituent shapes and shades so you can look at paint and see a picture.  
 
In my peripheral vision I see a bookshelf and the books in it. Except I don't. The acuity with which I think I see the world is available only in a few degrees of arc in the middle of my field of vision. Because I know the bookshelf is there, and because I know what a bookshelf looks like, my visual cortex fills in the details. I can directly gather only a little of my surroundings at a time. My brain fills in the rest so the world makes sense. 
 
Improvisers have three lines, at the top of a scene, to establish who they are; where; and what they are doing. From those three lines, the audience must understand — see — that the characters share an extensive context, and that this is the next moment in that context. This is impossible. 
 
Further: Before, or even after, the first of those lines, the players usually have no idea who or where they are, or what they are doing. When an actor utters the first line, the other actors have not agreed on an answer to those questions. He himself may offer that first line without any notion of what scene will come of it. Two lines later, the setting, the characters, and their immediate business are to be set in the minds of every player in the scene. This is impossible. 
 
And yet, it seems, it happens all the time (or at least half the time). The players toss out opening lines in near-desperation, and the audience really does see unique people who share a history. The audience sees this for the same reason they can look at a gray crescent and see a circle: It's not so much that they want the events on-stage to make sense; they can't help making sense of what they are seeing. 
 
The hardest thing for a new improviser to believe is that saying anything, promptly and with conviction, is better than waiting to craft a "good" line.  
 
Prospero I hate fish. 
 
Bang. First line. Do you know where this is going? I didn't when I wrote it. 
 
Salviati Hate him all you want, but Fish is a good cop, and I'm not taking him off this case.
or
Salviati Mr. Johnson, you hired me to organize a five-course dinner. Fish is the traditional first course.
or
Salviati So do I. We must destroy them all. 
 
Bang. The players are detective and lieutenant (or are they mayor and lieutenant?); if they are smart, they'll start talking about each other, or doing something in the office they've conjured up, to bring the focus back on-stage, and away from the absent Detective Fish. Or, the players are a social climber and the snob he hired to present a high-class entertainment; their relationship of master to servant is of opposite status from their relationship of novice to expert, and experienced players will make something of each character's efforts to uncross the relationship in his own favor. Or, the players are mad scientists of some sort; Prospero's job on the third line will be to establish whether he is colleague, master, or servant, and at least suggest where they are. They both have to work out why, by their own lights, they aren't crazy, because crazy characters might do anything, and the point of the exercise is to have actions come from character, not mental deficiency. 
 
To come back full circle: Improv players come naked, into a naked set. One offers a line — any line — and the other tries to make sense of it. Usually this succeeds, because brains are sense-making machines. The audience has sense-making machines of their own, and because they do they are naturally on the players' side. So long as the players give the audience enough support, keeping a consistent who, what, and where, the scene will make sense. The audience will, by necessity, fill out the remaining arc of the circle.