An 'offer' in improv is an initiation of a particular segment of interaction. It can be a question, accusation, proposition, or even a gesture, posture, or an odd look: anything that suggests it's the other player's turn in the interaction to respond and continue the dialogue. Offers are the building blocks of a good scene, but they are also fundamental to human interaction in general, not just in improv. By breaching the expectations of an offer -- in or out of a scene -- we can get unexpected, playful results.
So many of our conversations are structured predictably by what conversation analysts would call 'adjacency pairs,' which are prototypical offers and the expected responses that accompany them. For instance, if I say "My car broke down. Can you take me to the grocery store?" My preferred response is an affirmative "Oh, sure, I can do that." But even more broadly, my preferred response is some sort of answer, be it yes or no. Anything less would probably upset me.
To see how expectations structure day-to-day life, you can perform a breaching experiment to upset those expectations and see the subsequent confusion. In short, you usurp the 'serious' expectations of a conversational partner. For instance, when your roommate asks if you can take out the trash, you could pause and say "Shhh! It can hear us!" Then whisper like a hit man, "Listen, if you want me to take it out, it's gonna be double this time."
This will probably elicit one of a couple responses, including thinking you're off your rocker. Another one includes doubling down: "Sam, I'm serious, I've been taking the trash out every week." The transaction returns to the 'bona fide' mode and remains just that -- a transaction. There is no play. The play does not consist in the way in which the consequences add it; it comes from the total disregard for consequences!
Most people are too busy to stop and smell the roses, instead opting for the 'serious' series of consequences, the straight and narrow interaction, like an accountant stacking bricks on a scale. However, if your roommate puts on an Italian accent and responds "Sammie, the Don would not be very happy about double for the, err, trash. And you know how the Don gets when he's not very happy...", then we've got us a game.
A game is then defined largely by continuing the breach, even if the initial breach was accidental. Miles Davis once said that in jazz there is no such thing as playing the wrong note: it all depends on the note you play after. Similarly, Harvey Sacks discussed the ambiguity of first-pair parts in adjacency pairs (read: 'initiations'). There is no inherent 'preferred' response to an initiation; it depends on how that initiation is received: either seriously or non-seriously.
Since improv is not usually performed in your kitchen after dinner, there is a different series of embedded frames of expectation that comes along with the stage. To begin with, one should know from the get-go that the scenes bear no consequence on real life. And nor should they. There is never such a thing as a dud offer -- it all depends on how you receive it.
Play an offer like a slippery toad you toss back and forth, paying no heed to the consequences, foiling any attempt at total control, and you get an interesting, fun, and funny scene. Play the offer like a stack of bricks you have to respond to in order to balance the scale? Well, your scene might turn out to be a pile of trash.