Why is the swordfight in The Princess Bride so good?

Cary Elwes's As You Wish: Inconceivable Tales from the Making of The Princess Bride gives a lot of detail about why the sword fight turned out so well. Neither Elwes nor Mandy Patinkin had a sword-fighting background.

But they worked with excellent trainers:

Peter Diamond was a good three decades into what is generally regarded as one of the most legendary careers of any stuntman or stunt coordinator in both television and film. As a sword-trainer he had worked with both Errol Flynn and Burt Lancaster. And in the previous decade alone, he had served as stunt coordinator on the original Star Wars trilogy. For you “Wookieepedias” reading this, the Tusken Raider that surprises young Luke Skywalker on the Tatooine cliff top with that horrifying scream? That was Peter. He had also been the stunt arranger and coordinator on movies like From Russia With Love, Raiders of the Lost Ark, and Highlander. Classically trained at the Royal Academy of Dramatic Arts, Peter had also appeared in front of the camera, not only as a stuntman but sometimes as an actor as well. That’s him as the German soldier Indy notices in his side mirror climbing along the side of the eighty-mile-an-hour speeding truck without a harness in Raiders. Peter logged more than a thousand credits before passing away in 2004, at the age of seventy-five. He was vibrant and actively employed until the last year of his life.

Bob Anderson was also a native of England and also something of a national hero, having served in the Royal Marines during World War II and as a representative of Great Britain on the fencing team in the 1952 Summer Olympics in Helsinki. He later became president of the British Academy of Fencing and a coach for the British national team. His expertise as a swordsman eventually took him to Hollywood, where he became a sought-after stuntman and fight coordinator. The man’s résumé was breathtaking, from coaching Errol Flynn like Peter in the 1950s to choreographing fight scenes for several James Bond films in the 1960s, and working alongside Peter in From Russia With Love and (Star Warrior alert) on the Star Wars trilogy. That is Bob using the dark side of the Force as Vader in all the light-saber sequences. Bob also passed away, in 2012, at the age of ninety, but worked until the last, serving as “sword master” for Peter Jackson’s Lord of the Rings trilogy.

Elwes, Cary; Layden, Joe. As You Wish: Inconceivable Tales from the Making of The Princess Bride (Kindle Locations 927-942). Touchstone. Kindle Edition.

He gives a sidebar to Mandy Patinkin, who talks about the training he did even before he and Elwes began rehearsing together:

Goldman wrote in the introduction to my character that he is “the world’s greatest sword fighter,” and I figured, that’s what I’ve got to learn how to do. So I immediately got in touch with Henry Harutunian, who was the Yale fencing coach, and we worked together for two months. He taught me the basics of fencing. I was a righty, and he taught me first how to fence with my left hand; we worked the left before the right, and I actually became a better left-handed fencer than a right-handed fencer.

Elwes, Cary; Layden, Joe. As You Wish: Inconceivable Tales from the Making of The Princess Bride (Kindle Locations 948-951). Touchstone. Kindle Edition.

They started out practicing forty hours a week, and in pretty much any spare moments during the production, they were practicing.

So the answer is: Rob Reiner hired the best in the business to train Patinkin and Elwes, and they put in a lot of time learning the techniques.

Other answers have focused on the technical side of the fencing. As you'll no doubt find on YouTube breakdowns, a number of elements of this are clearly not pure combat. Most obviously, if they genuinely intended to kill each other then they'd probably both start off using their right hands! No, this is stage fighting - but it's very very good stage fighting.

The point of stage fighting isn't simply for the character to win or lose, but for the manner in which they win, lose or draw to demonstrate the personality of that character. When a lead character is fighting mooks, of course they'll just block the initial swing, stick them or slice them and move on, demonstrating that they're just that much better than a regular opponent. But fighting another lead character, how they both fight has to tell a story about both characters.

The scene starts with a mutual admiration session, demonstrating that they're both good guys. The interesting thing this does is that we're no longer rooting for one or other of them to win - we like both of them, and we don't want one of them to drop out of the film. This is easier to do with unarmed martial arts, where one character simply wakes up with a headache, but it's much harder when there's swords involved.

Then we watch how the duel develops. With that mutual respect, neither character is outright trying to kill the other - rather they're clearly feeling out their opponent, manoeuvring for position, and seeing what the other person does. That develops through various rounds of disarming, changing hands, gymnastics, and using the terrain in interesting ways, and the action becomes progressively more intense. Where this is unusual is their comments on each other's abilities, and again this changes the tone of the scene from a battle to the death into something which is actually supposed to be fun.

Up until the gymnastics though, it really isn't clear who has the edge, and that sustains the tension through the scene. The flips off the bar are the point where it becomes clear that Westley is a little bit better than Montoya. From that point the fight reaches a satisfying conclusion which is consistent with what it's demonstrated about both fighters, based on both their levels of skill and not on some badly-scripted accident.

The fencing certainly is beautifully done technically. The balance, movement, distancing and sheer ability is wonderful to watch. What makes this a great film scene though is the story it tells.

Perhaps not sleeping on the couch, but they did employ the guy considered to be the best in the business.

From Wikipedia - Bob Anderson

Robert James Gilbert Anderson (15 September 1922 – 1 January 2012) was an English Olympic fencer and a renowned film fight choreographer, with a cinema career that spanned more than 50 years and included films such as Highlander, The Princess Bride, The Mask of Zorro, The Lord of the Rings, and Die Another Day. He was regarded as the premier choreographer of Hollywood sword-fighting, and during his career he coached many actors in swordsmanship, including Errol Flynn, Sean Connery, Antonio Banderas, Viggo Mortensen, Adrian Paul, and Johnny Depp. He also appeared as a stunt double for Darth Vader's lightsaber battles in The Empire Strikes Back and Return of the Jedi.

In preparation for the fight, Patinkin & Elwes spent months rehearsing for the scene, learning to fence both right and left-handed.

Refs:
Wikipedia - The Princess Bride
CinemaBlend - Why The Princess Bride's Epic Sword Fighting Scene Was So Hard To Make

As mentioned it took lots of training, and Elwes' behind-the-scenes provides more detail into the monumental effort that may be less visible, but incredibly important to support the leading actors.

Source material

Goldman wrote the "Greatest Swordfight in Modern Times" using sources to inform his vision (p.66):

Books like The Academy of the Sword (1630) by the Flemish master Gerard Thibault d'Anvers. Or Great Representation of the Art and Use of Fencing, written by the Italian maestro Ridolfo Capo Ferro and dating back to 1610. And even Treatise on the Science of Arms with Philosophical Dialogue

Athleticism

The pairing of Elwes and Patinkin created competitive atmosphere. As Reiner describes (p.70):

I'm sure there was a sense of competition between Cary and Mandy, and I think that was probably healthy. This is a duel to the death, supposedly, and so it is a competition. I think that was there, for sure.

When trainers Anderson and Diamond, set expectations about how much can be learned and the potential need of stunt doubles for the "key moment in the sequence" (p.82):

Mandy, who said, almost without hesitation, "Don't worry. We'll get it."

The room was quiet for a moment. Mandy looked at me. What was I going to do? Did I think a stunt double might be necessary to stage "the Greatest Swordfight in Modern Times"? Of course it occurred to me that it might not be a bad idea after Bob had suggested it. Was I going to admit that now, seconds after Mandy had promised no such assistance was necessary?

Inspiration

The first exhibition of Elwes and Patinkin swordfighting on set (after basic training and the first couple months of shooting), they finish to applause by the crew, Anderson, and Diamond. However, Reiner responds "That's it?" (p.199):

Not exactly the response we had anticipated, as I'm sure you would agree. Mandy and I had spent so many hours practicing and perfecting the duel, mapping each and every step of the choreography, every thrust and parry of the fight, that we were now able to perform it not only fluidly but flawlessly

As a group they brainstormed,

collect every single swashbuckling movie available on video, including the ones we had already watched, and watch them again to find what we needed. Movies like The Crimson Pirate, The Mark of Zorro, Captain Blood, The Black Pirate, Adventures of Don Juan, The Count of Monte Cristo, The Three Musketeers, The Scarlet Pimpernel, The Sea Hawk, The Prisoner of Zenda, Scaramouche, etc.

Patinkin is quoted (p.207)

Rob wanted the actors to be seen doing all the fencing. He wanted full-body shots, as opposed to most other fencing pictures, where it would be the point of view of the actors. Where you would see only the hand of the other fencer off camera. In most movies, this would be done by a stunt double. But Rob was adamant that we do all the fighting ourselves.

It's really staggering, their on screen achievement is a testament to their entire production. From changing the set to build the tower and steps by production designer Norman Garwood (Time Bandits, Brazil, The Missionary), down to perfect fitting wardrobe by costume designer from Lawrence of Arabia and Doctor Zhivago, Phyllis Dalton.