Until the early 18th Century, fencing was the preferred form of prizefighting in England. Prior to the English Civil Wars, prizefights were for all intents and purposes the examinations for Masters of Arms. If you wished to teach the use of swords (broadsword, rapier, two-handed sword, sword and shield, etc.) as well as other weapons (quarter staff, halberd, etc) you had to fight established masters in each of these weapons.

These prizefights were incredibly skilful, and the weapons were sharp, so things also got quite bloody. The blood, plus the fact that the examinations were held in public, made them insanely popular. Crowds would throw money on the fighting stage if they found the spectacle sufficiently violent/entertaining, and the candidate, presuming he passed his tests, would use this money to start his new fencing school. Presumably the skills of master-level combatants were such that they could give an abundance of crowd-pleasing bloody wounds without accidentally killing each other.

Following the Civil Wars, the prizefights lost the formal examination aspect, and devolved into a pure spectacle, a sort of gladiatorial entertainment, or martial arts vaudeville. Popularity remained high, and in the early 18th Century one expert fencer and promoter, by the name of James Figg, got the idea to introduce warm-up acts into the show to get the crowds ready for the main spectacle. The warm-up acts were bare-knuckle boxing fights.

In time the boxing fights became more popular than the fencing, and sword fighting as popular entertainment went the way of the dodo. Prizefighting became synonymous with fist fighting, and James Figg went down in history as a famous boxer--in fact the first boxing champion--when he was actually a fencer by trade.

That doesn't necessarily explain why boxing has held on to this preeminent position in the world of entertainment combat sports for so long, but it's interesting to note that MMA-style fighting was never a real competitor against it, at least not since the ancient Olympics and pankration.

Read more: The English Master of Arms by James Aylward.