There are many different facets of improvisation (the art form) that apply to RPGs and can be used by players and DMs alike. Some parts are more useful than others, and actually some parts will even seem to run counter to particular styles of Dungeon Mastering. At the end of the day, studying, practicing, and eventually hopefully mastering improvisation techniques will just add tools to your tool box if you choose to go in that direction. Not necessary, not a requirement, but definitely can make you a more comfortable player if you embrace them.
The biggest advantage is that it just makes you more comfortable making choices. In improv since there is no script whatsoever, the only way a scene can start and go forward is by someone making a choice and verbalizing it. You have to make a choice (any choice)-- whether that's a statement of who you are, how you are feeling, where you are, what you are doing, anything. You establish something and it is now true. It doesn't matter whether its good for you, bad for you, a risk, a walk in the park-- that stuff doesn't matter. All that matters is that you make a choice and impacted your reality. And just like in RPGs this is crucial. Your character in D&D has to make choices-- the story and the game do not progress if you don't. And while some players can get away with letting the DM push their character forward and make choices for them, becoming more comfortable making your own choices will just speed your game up, give you agency, and allow you to use your imagination more fully. The game will generally just be more fun if you allow yourself to make choices and see what results from them (good or bad). And improvisation teaches you that good or bad doesn't matter-- because you will always make another choice in another three seconds that will more you forward and perhaps turn you in another direction, so there's no reason to fear making a choice.
Improvisation is also wonderful in preparing you for being able to react to the unexpected. Indeed, the entirety of improvisation *is* reacting to unexpected circumstances, because you have no script, no dialogue, nothing established beforehand. You have no choice BUT to react. Sometimes you get to react to favorable circumstances, sometimes you need to react to really bad circumstances. But once you've done it enough times, you become inured to it. As a player or performer, it no longer panics you when something "bad" has happened. Whether that's within the story itself, or even "out-of-game". All these things are just new choices that have been offered to you, and you react to them as honestly and in-character as you can. Which is why you'll sometimes hear people mention in improv that "there are no mistakes". And there's a reason for that-- if you just react to everything that happens as it happens, whatever "mistake" occurred will just get incorporated into what the story is.
In RPGs this is a fantastic tool for a DM. Because you no longer "sweat the small stuff". If you've thrown an encounter at your players and they absolutely decimate it... no big deal. You react to it. Is this encounter meant to be a throw-away? Then you let them have their easy victory. Was it meant to be a climactic battle? Then F-it... throw in another wave of monsters. Or have an earthquake start bringing the caverns down on their head. Or come up with something else. But at the very least, you won't panic that the encounter didn't turn out the way you were expecting, which would only result in you either getting defensive and upset, or you start making choices without consideration in an attempt to "save" the situation (and will be much more obvious to the players as "ploys" to save the DM's precious story and will be seen as possibly "unfair" tactics).
Being comfortable with reacting results in many fewer "oh SH!%$!" moments on the part of the DM, and thus anything you do will come off as just a regular part of the story being told and make it much more palatable to the players (since in truth they will have had no idea that what just occurred wasn't meant to occur all along.)
And finally there's also just (as was mentioned above) the team-building and cooperation aspect to improvisation and the game. You are all in this together. If you want the scene or the game to work, you need to listen to one another, support one another, be comfortable with one another, respect one another. In improv-as-performance you HAVE to do this because there's an audience out there watching you do it. And if you can't trust your fellow performers, then you shouldn't expect a high batting average of success. And likewise at the game table... if you can't expect your wacky choices to be supported by your teammates (and if you don't support your teammates' wacky choice either), then the very fabric of the game will slowly unravel. It might take several months, or it could occur in just several sessions... but you'll be able to sense it right away when it's not coming together. And at least by having trained in improvisation you will have learned a better sense of what will and won't work so that at least YOU weren't the person that caused the unraveling in the first place.
At the end of the day, the biggest detriment to both improv and D&D is fear. Fear of screwing up. Fear of disappointing people. Fear of looking bad. And performers and players will throw up roadblocks all along the way in order to shield themselves from the things that cause that fear. But what you'll eventually find is that those very roadblocks you put up aren't shielding you from the things you fear... they are directly responsible for them. Because those roadblocks are what keep you from making strong choices, and trusting your fellow players to react to them... and them then making choices and you then reacting off of them.
If you can get past your fear, then a whole new world will open up. And studying improv is a fantastic way of doing that.
