The primary purpose of the Monsters is fairly simple: They seem to act as a source of post-scarcity peril and deviancy, specifically designed for the strangely filtered lifestyles of Elysium.
It isn't known who or what created the Monsters. Presumably, they have existed on Elysium as long as the island itself. Rumors and legends have long referred to a shadowy group of "Forefathers" that created Elysium and its environs, and some have started attributing the Monsters' presence to them as well. That the Monsters show no knowledge of the Mess beyond the shores of Elysium helps support this hypothesis.
Observance of Monster behavior patterns by RBAC and other unaffiliated VRC researchers shows that they seem to act as scapegoats for the Elysians, acting as a sort of "safety valve" by satisfying the islanders' repressed sexual deviances and urges in a private setting away from the Elysium population. They seem to be able to do this regardless of who their "victim" is, a trait that points to some sort of "sixth sense" that can read a person's unspoken desires so they can be acted upon. Outright telepathy has not been ruled out, as there are recorded instances of Monsters knowing other things that the victim thought were private and secret. Some researchers have theorized that the Monsters may harbor other engineered physical and mental abilities on the assumption that the Monsters were indeed purposely created, but none have yet dared brave the conditions necessary for continued experimentation.
Only two Monsters have awoken so far, a dragon and a minotaur that can both be found on the eastern side of Elysium. Guarded whispers and wild rumors have speculated on the existence of others slumbering on the island, simply waiting for the right moment to emerge...
There's much more to the Monsters than is revealed here. We leave it as an exercise to the characters in play to discover the rest. ;>