The Mirror Bosses in Dragon Quest Swords for the most part are just bosses that were added into US and PAL copies of the game.
However, there may be an actual story involved. The fights with the Mirror Bosses all take place in an alternate version of Avalonia, which already implies that something had happened. Considering that we never see any other characters other than the bosses, it's possible that all of them were killed by the monsters that now roam the alternate Avalonia.
Each of the bosses, for the most part are stronger versions of monsters seen in the main story. The first one is King Latem (Metal King) (considering he's fought near the item shop and The Hero's house) who doesn't attack at all and will flee after 25 seconds.
The Mirror Bosses after that are fought in the town centre, with the castle behind them. Nomeg (Gemon), Salta (Atlas) and Clank 'n' Knalc (Clank 'n' Knalc) are the first 3 monsters to fight you, two of whom are mirror counterparts to Gemon and Altas.
After The Hero slays them, the player comes face to face with Valgirt (Triglav), who is capable of reviving itself once. The Purple Mirror bosses Nomegoen (Neo Gemon), Der Gib (Big Red) and Valgirt Nedlog (Golden Trigalv) are all stronger versions of 3 other mirror bosses, implying that someone had resurrected them...
This is further supported when the player has to refight Nomegoen, Der Gib and Valgirt Nedlog in sequence before fighting Edahs 'Sohpix. It's possible that this sequence is what happened canonically, if the Mirror Bosses were canon, assuming that Edahs didn't just revive them one last time.
Now the most interesting part is Edahs 'Sohpix (Xipho's Shade). Considering how much stronger he is than his main counterpart, it's possible that Edahs 'Sohpix killed The Hero and then laid waste to Avalonia. As such, alternate Avalonia is what would happen if you fail to defeat Xiphos. It's possible that the player fights Edahs in Avalonia Castle, considering that you had to mow down 3 of his elite monsters just to get at him (even though his battle arena is the same as the final boss).
Of course, with all this in mind, it's possible that Payback Mode is the alternate Hero's second chance to stop Xiphos before he lays waste to the world. This time, The Hero is now armed with Xiphos/Edahs's sword, and can use it's dark powers to get payback for his death.
All of this of course is a theory, and isn't official.