In the movie Predator (1987), why did the predator come to Earth? I only know a little bit about Predator lore/fandom. Was he hunting for a rite of passage? Was he already an established hunter and just hunting for more trophies? (These are just guesses of mine)
The short answer, which is probably the best answer, is that he was an experienced hunter and wanted to hunt "the most dangerous game"; a combat soldier. You'll note that he never kills unarmed humans (which is why Anna got sent to the chopper unarmed), and at the conclusion he decided to engage in hand-to-hand combat against Dutch, whom he deemed to be a worthy opponent.
In a 2017 interview with all the Predator key actors by Pete Keeley for The Hollywood Reporter, Jim Thomas (who wrote the script together with his brother John) said:
But the original conceit was always, "What would it be like to be hunted by a dilettante hunter from another planet the way we hunt big game in Africa?" And at first, we were thinking about how a band of hunters would branch out and hunt various and dangerous species on the planet, but we said: "That's going to be way too complex." So, what's the most dangerous creature? Man. And what're the most dangerous men? Combat soldiers. At that time, we were doing lots of operations in Central America, so that's where we set it.
Predator culture revolves around the hunting and stalking of dangerous lifeforms. After making a kill, Predators typically skin or decapitate the carcass, converting it into a trophy. If immobilized or at the brink of death, a hunter will activate the mass-explosive self-destruct-mechanism in his wristband, honorably erasing any trace of its presence to its prey. It is often alluded to that the reason Predators hunt is not for sustenance or elimination of threats, but as sportsmanship or rite of passage, as they will normally only attack life forms that have the ability to provide them with a challenge. In Predators, it is revealed that there are at least two different Predator tribes, which are engaged in a long lasting blood feud. The film also introduced a pack of spined, quadrupedal beasts used as flushing dogs by the "Super Predators," Creature designer Gregory Nicotero used hyenas as a basis for the creature's physique and the spines were added later by Chris Olivia.
Predators made contact with early human civilizations such as the Ancient Egyptians, the Khmer Empire, and Aztecs, as well as a fictitious culture inhabiting what is now Bouvetøya. Upon arriving on Earth, the Predators were worshipped as gods by humans, and they taught many of the civilizations how to build pyramids (an explanation as to why many of these different ancient societies had distinctly similar cultures and architecture), but in return expected sacrifices of humans for use as hosts for huntable Xenomorphs (Aliens) – the ultimate prey for initiates. The Predators returned to Bouvetøya every century to consummate the bargain, until at one point in the ritual, the Xenomorphs spread out of control, resulting in the Predators detonating a bomb that obliterated the entire civilization. Relations between humans and Predators deteriorated from that time on; the Predators then viewed humans as little more than another quarry to hunt.
Predators feature prominently in the folklore of certain cultures; some Latin American people refer to the species as "El Diablo que hace trofeos de los hombres" (Spanish for "The Demon who makes trophies of men"), and Jamaican superstition identifies Predators as demons from the spirit world. When hunting humans, Predators normally avoid certain individuals such as children and some adults if they are unarmed, though they will spare armed ones if they happen to be pregnant or sickly unless they are attacked by them. A human who has managed to kill a Predator or a Xenomorph in single combat or has fought alongside a Predator is usually spared by the deceased hunter's comrades and given a gift (often a rare or exotic weapon) as a sign of respect.
Much of the source here is the director's commentary. Basically, the Predator species seek out warlike races and use them as sporting prey. Humans are pretty good at killing each other, so....