I don't think I'll ever be brave enough to take improv classes, but I've absorbed a bit about the subject through cultural osmosis. I think we've all become acquainted with the phrase "Yes, and..."
There will come about a million times when you are gamemastering that a plater will do something you did not expect. They will ask a question you did not account for in your adventures prep or head south instead of north or want to have an in-depth conversation about the local flora with a herbologist NPSC that you had not created. In these cases you will have to improvise.
This is a skill that I think all GMs develop over their years of experience. You may develop a stable of NPCs that you use from game to game to fill the most urgent vacancies- this may then lead to your players jopking about how many retired soldiers turned barkeeps with eyepatches named Baris or Laris or Daris exist in your games - but you can always remind them that cliches become cliches for good reason.
Until you develop this pool of "go to" characters, events, locations, etc. you could use one of the gamemaster's oldest friends: a table full of stuff that you can roll on - covering everything from names, mannerisms, hair color, and innermost secret desires for your NPCs for instance. These tables are plentiful in most GM guides.
There are moments of spontaneous worldbuilding or plot fabrication, however, where a player writes you a blank check by asking for a specific thing:
PLAYER: "Is there a secret high-stakes card game I can get in on in this city tonight?"
"Is there an office for that caravan company we helped a few adventures ago nearby?"
"Could I find a place in town to set up a soapbox and preach for a while?"
In these instance you are probably expected to answer with a yes or no, but you should take a moment to consider what each option means for your game. Then consider how you can take what is probably a suggestion of what the player wants to see in the game and make it work for your game. Even if you cannot answer a player's question with a "Yes, and..." you may be able to answer with a "No, but..." that provides a different sort of satisfaction.
So if the question is:
"Is there a secret high-stakes card game I can get in on in this city tonight?"
You might consider these answers and their implications:
YES: This could mean that the city is large enough to hide criminal activity or that there's an established gambling underground here. This could mean that the law enforcement is incompetent or receptive to bribes. This could mean that you can find ANYTHING in this city.
NO: This could mean that the law or the people of the city would not tolerate such a thing. This could mean that most everyone in the city is too poor for high-stakes gambling.
You might say:
"Yes, and you've heard that the game is run by a particularly cutthroat gang called the King's Street Brigands."
"Yes, and you've heard that even the Mayor themselves gets in on the action from time to time."
"Yes, and you'll come to find out that they play for more than just coin..."
"No, but you know that is because this town takes a finger from convicted gamblers."
"No, but you doubt that anyone has two coppers to rub together- much less gamble away."
"No, but you think you might find a more casual game if you crawl the taverns near the dock."
Any of the above suggest something that you may not have previously thought of regarding this city or gambling in your world. This makes for great jumping off points for developing characters, storylines, and content. You may eventually reach the point where you reply with a "Yes, and..." or a "No, but..." statement so quickly and seamlessly that your players aren't sure if you had prepared for their question or not.
That's what I would call solid GMing skill...
