I watched action movies, Bruce Lee's movies, Steven Seagal's movies and realize that while fighting, they (or their enemy) just stop punching and stand and sustain the other's attack. Why is that?
I don't know about the movies, the reason has more to do with the plot than an actual answer.
In the real world, people exhibit one or more of six responses to an attack: fight, flight, freeze, negotiate, capitulate, and contemplate.
In your question, those who "just stand there" are either contemplating their move, which isn't an unreasonable response if the victim was not yet being struck and had time to ponder options; or could be freezing, an subconscious result of fear and not knowing what to do.
But in the movies, the tactic is more likely a result of the antagonist trying to scare the protagonist by making the protagonist think the antagonist has no fear. If the antagonist is not affected by the protagonist's strikes, this is more likely to be effective. I'm thinking this was the case in the movie "The Punisher" when Frank Castle was fighting The Russian. The Russian was a huge monster who was impervious to anything that could be delivered (except for a pot of hot water, LOL!) and in several strikes just stood there when Castle delivered several (ineffective) blows. Rope-a-dope, as Muhammad Ali was famous for using, was also used by Rocky Balboa in several of the movies, but rope-a-dope isn't quite the same as the behavior you describe.
It's basically for dramatic effect.
In any fight be it competition or self defence or anything else you never stand still. This is to showing to the audience that they can block without moving and so must be much better at fighting than whoever they are facing... which is the point of most of these action films.
Sometimes it also drives the plot as in some films they want to show that the hero has some other motive than killing his opponent.
In either case it has little to do with regular martial arts teaching or with self defence and everything about showmanship. Stage martial arts are a skill in themselves but only sometimes transferable to actually fighting.
Sadly, this happens in most martial arts schools in real life, too. The instructor typically has his partner do something to him, and his partner will just stop immediately afterward. Then the instructor does a bunch of other things to him in rapid succession.
The purpose is obvious. It's to make the instructor look like a martial arts god. The same thing is done in movies for the same reason. It's to make the main character look super-human.
By the way, I want to clear something up. This is something that actually took me a while to figure out and to solidify in my brain. So I want to pass it on:
Whenever you see anyone stop after doing something, and his instructor is just doing a bunch of things to him in rapid succession, that's fraud.
It's okay to demonstrate what to do immediately after the punch comes, for example. But instructors cross the line when they continue doing a bunch of other things after that to an opponent who's just letting him do it. When they do that, it's just fraud. And it's BS.
The only time it's acceptable is in movies, or for entertainment purposes.
If you belong to a school where this happens a lot, especially if it's the primary way you're being taught, then it's time to wake up and realize that you're being misled. The stuff you're learning doesn't work like it's being taught. It probably wouldn't work at all. And in the end, your false confidence might get you in a lot of trouble should you try using it for real.
Just wanted to pass that on.
Without knowing the exact situations you're citing, one possibility is that someone is doing a "rope-a-dope", using a strong defense and letting their opponent wear themselves out before starting their counter-attack. Fighting is hard, and someone constantly on the attack can wear out quickly. If you're in a situation where you can minimize damage (the classic boxing case is either just barely dodging out of the way, or backing into the ropes and letting the ropes take some of the force of the blows as you block), it might be worthwhile to let your opponent wear down and then make a comeback. If you're not actually in the midst of the combat, and just standing there while your opponent circles you, you're also potentially coming out ahead on energy spent.
If it's a brief pause, they may be attempting to throw off their opponent's rhythm to create an opening. Fighting is about patterns, and disrupting the pattern can be effective. if it's a longer one, they might be more confident in their counter-attacking skills. By forcing the other person to approach, they're controlling the situation. In addition, it dovetails with what Huw Evans stated above about superior ability. Someone willing to stand still and act is suggesting they don't need to move, which could provide a psychological edge if they convince the other person that they're just that good.
And yes, some of it will be cinematics. It makes a person look good if they don't even have to move to fight. It makes the camera work easier. And it avoids some potentially fight-ending accidental injuries like sidestepping the attack and putting your foot down on a slick spot on the floor, or a prop, and turning an ankle.
I suppose the quickest or shortest answer (IMO), is that you don't want to waste YOUR energy while your opponent seems to be perfectly fine with wearing themselves out, wasting their own energy. The "What-Ifs" are numerous. They could have friends who may want to fight with you, your opponent could grab/find a weapon to all of a sudden tip the odds in their favor a little bit, or a huge amount. So, maybe an even better (and shorter) answer is so that YOU can remain "ready for anything" that is within your control. - Michael