Actually, by the manual of the game you are not supposed to fall down but run back to the surface. Yes, you play entire game, only without big bosses, in backward direction. The game mechanics is that you have the same amount of time to escape back as much as you spent to reach the main villain. Once you killed him, you started the volcano which will destroy the underground kingdom, so you have to escape on time.
So, Hegor doesn't automatically jumps, it's the player's choice, and you can see player used "move left" command. Probably they figuered out that falling down gives you Congrats message, which dying on the way back doesn't. So it's probably bug in the DOS version.
Here is the video of same uploader where he tries to get back to surface and dies. Look for the Amiga version longplay and you'll see the erupting volcano at the end.
Plot is: hero Hegor is son of Thoron. The main villain is Thoron's twin brother Necron. When twins were young, druid trains them combat skill, shows them they have magical powers and prophesizes that good will triumph. Necron had ability to shoot fireballs with his hands and was given a bow, probably the one you find in game. He is embodiment of evil and becomes a tyrant. Thoron is embodiment of good and is given a sword, the one Hegor carries. Thoron is gifted with some kind of defensive magical shield (well he's a good guy) that can reflect Necron's fire, so he is the only one that can stop Necron. A lot like cartoon Wizards if you saw it.
So Necron becomes bad and Thoron fights him without either using magic as it is useless against one another. Thoron beheads Necron who then turns into energy ball. Thoron somehow knows he must travel to Mordor and throw it into the lava... no wait, he just knows fires from the center of Earth can destroy such energy ball but can't do it since there is no lava as they were fighting in the town and he feels sorry for his brother. Insteads he is satisfied that his brother can't do any evil anymore. When energyball saw its brother won't kill it, it transports itself and the bow to underground, where necromansing Necron creates his undergroun kingdom.
Thoron goes to nearby village to settle down. When hellhounds kill his wife, he realizes his brother is alive and wants to kill his family not just him. So he doesn't take new wife but teaches his son Hegor to be warrior. Hegor witnesses his father defending the village from a dragon and gets eaten, the same dragon you face in the game, Necron's guardian. Hegor goes around slaying dragons, getting drunk and getting laid (really) until Necron's power is so strong to start havoc in the land. King will give the crown to anyone who can stop him, Hegor goes back to home village and his father's spirit in disguise tells him the full story and takes him to marshlands so good can finally triumph. Fathers...
So, right in the plot it is suggested how to defeat Necron, although Hegor doesn't have this power like his father, but must find "means" to deflect fire.
And that's it. After defeating evil and escaping volcano, Hegor got bored of life as king and went off to chase Red Sonja, eventually spending all his money gambling in taverns. So broke but still famous, he tried to have acting career, appearing in lot of B-movies always starring as blonde rough barbarian. He scored one big role though, in a movie partially inspired by his adventures about a lava killing a guy called Nekron. Ralph Bakshi gathered star crew from fantasy B-class video-games, trying to use notable actors from game flops due to increasing popularity of fantasy movies. The only character from successful game was the girl from the other Barbarian game. She and Hegor were cast as leads and protagonist from 1989 C64 game Storm warrior played the Darkwolf. But it was a flop although awesome cartoon, gaining cult following and nailing most of actors careers to new game flops and B-roles.
Another interesting thing: guy playing Darkwolf actually gained lot of popularity with the movie and got big production offers, but wanted to move away from fantasy to SciFi hoping for better commercial success. He starred at AAA game CyberMage - Darklight awakening - you guessed it, he was fighting Necron again. Again game was a flop although technically on excellet level. He appeared in one successful title - I think it was him playing mute alien in PC game Another World.