Welcome to the world of Pokemon!
Prehistory
A million years ago, a tiny, highly dense, highly radioactive meteor hit the planet somewhere in the Hoenn region. Having the size of a soccer ball but the mass of a house, it sank through the dirt like water.
Its incredible radiation spread through the planet. The animals and plants of earth adapted to become far more resistant to it.
However, rapid mutations began to occur, sometimes among humans. In just a couple dozen generations, there were lizards that used exothermic biochemical reactions to spit fire, giant tortoises, dinosaur-like monsters that allowed plants to grow on their backs in a symbiotic relationship, animals that evolved a super hard shell that looked exactly like rock or steel for camouflage, even unicellular organisms that created gas around them to give the appearance of a ghost.
The radiation of the meteor was no ordinary radiation. It had a seemingly unlimited source of power, and it could be used by the monsters to make themselves stronger. Some monsters had even learned to use that power to perform "psychic"-like feats: such as reading the brainwaves of nearby creatures, creating gravitational fields, or even using the power as a tangible beam of energy.
It has also been said that there are "legendary" and "mythical" Pokemon with even more incredible abilities, such as giving life, manipulating time, space, land and sea, but there has never been a proven sighting, capture, or even photograph. They are called "legendary" and "mythical" for a reason: they most likely exist only in legend and myth.
These monsters would battle amongst each other for local supremacy. It eventually came to be that all monsters in one area would be around the same strength level, as the weaker ones would be eaten and the stronger ones would be defeated by the very strongest monster in the area intending to keep its title.
After battle, monsters would rapidly mutate due to the radiation of the meteor that built up in their systems after battle. These mutations could also be caused by the radiation of stones that were affected by the meteor and brought to the surface by earthquakes. They could be made stronger, allowed to learn new abilities, or even metamorphosed into totally different, stronger monsters.
Needless to say, these monsters were far stronger than humans. Although only a few species could eat humans, and even fewer were ever hostile towards them, humans nonetheless held every one of them in the same regard: as fearsome, vicious, monstrous, creatures. The other species that had remained unmutated were driven into extinction by the monsters. Humans frequently hunted the young, old and sick monsters, or even destroyed their habitats. The monsters became as scared of humans as they were of them. Humans leveled clearings into forests and carved cliffs into mountains to build cities where the monsters would hopefully be too scared to go, but often cities were destroyed in attacks.
It was very, very hard to travel between these cities. This fact created two important aspects of the culture: One, if a weary traveler knocked on your door, you would be obliged to provide them with a hot meal, a place to stay for free, and perhaps a gift, for you yourself might end up traveling and in need of a place to sleep before you know it. Two, each city only had the natural resources available nearby because trading was almost out of the question, so technology progressed at an extremely slow rate.
Due to that very slow moving technology, monsters even began evolving to camouflage among man-made objects.
The first trainers
One day, hundreds of thousands of years ago, somewhere in the Kanto region, humans attempted to capture a monster for use as a weapon against other monsters. This was revolutionary. The only thing strong enough to defeat a monster was another monster, or so was the logic of these early trainers. They found that stealing and incubating an egg was the easiest method, because the newborn monster would imprint on the stealer.
These people would train their monsters by forcing them to fight wild monsters or other people's monsters. With strong enough monsters, one could finally travel safely. The gap between cities shrank.
The radiation, for some reason, built up more quickly when a monster had a trainer. As an emotional bond grew between them, more and more power would accumulate inside the monster.
One day, someone discovered shards of a second meteor that had fallen to the planet, but had shattered in the atmosphere this time. It was discovered that these stones could be used to harness the power of the first meteor, built by the bond between trainer and monster, in order to transform the monster into a more powerful version of itself. This was kept secret for many years.
Between the regions, a battling system was universally agreed upon: if you lost, you paid the winner. And you couldn't refuse because the winner still had a battling monster that could easily pulverise you. Instead of working together to make their monsters collectively stronger, which would be mutually beneficial, the trainers wanted to shove their way to the top. The hostile ecosystem had created an every-man-for-himself mindset.
Quickly, the absolute strongest trainer rose to leadership in each town. They would build a gym: a place of training, and anyone could go there to challenge them for leadership.
In an entire region, the number one strongest trainer was sovereign leader. To challenge for that position, you needed to prove yourself in a series of trials. First, you had to travel around the region; which in itself was hard, defeat a minimum of eight gym leaders, collecting their badge to prove your victory, then travel through a "victory road," a place where the most powerful monsters dwelled.
When you finally made it to the "League," which was deliberately built where those strongest monsters lived, you had to defeat the "Elite Four" all in one sitting.
The "Elite Four" were four strong trainers, each specializing in one "type" of monster. The position was often hereditary.
After doing all that, one could finally challenge the champion for reign of the entire country. This entire system evolved into being, first perfected in the Kanto region, and then mimicked by other regions.
Given the dangerous nature of this profession, only the strongest were able to kidnap a monster to train. This created a mindset that monsters were weapons of war and nothing else.
The Pokeball
One day, about a hundred thousand years ago, someone perfected the Pocket Ball. It had been only a theory for years, but eventually someone got it right. Upon striking a wild monster, it would shrink the monster to the size of the ball and encapsulate it inside. It was a monster's natural instinct to curl into a ball when resting or injured, and that was the basis of the design. Also, it was designed to seek the radiation of the meteor, so that monsters would be the only thing able to be captured.
This was accomplished by slightly shrinking the nucleus and slightly growing the electrons inside the atoms of the monster. The electrons would orbit closer to the nucleus, causing the atoms, and thus the monster, to shrink to the size of one's pocket. The ball also put the monster in almost zero gravity so it would weigh nearly nothing.
Of course, Pocket Balls were at first ridiculously expensive, but they were mass produced and shipped until gradually they became as cheap as water bottles.
"Pocket Ball" was quickly shortened to "Poke Ball." Monsters became known as "Pocket Monsters," because they could now be stored in your pocket, or "Poke Mon."
Now, anyone could train any Pokemon, if they could catch it. Pokemon training was no longer reserved for the most able people. Even small children could train Pokemon. The image of Pokemon as fearsome weapons of war was, for the most part, replaced with that of friends, allies, faithful servants. This is evident when one compares artwork of Pokemon before the invention of Pokeballs and after.
League Regulations
The invention of Pokeballs caused the League to make a few rules. In order to train Pokemon, you needed a license, which you could not get until you were a given age, anywhere between ten and sixteen depending on the region.
A trainer could only use one Pokemon in battle at a time, unless it was a "double" or "triple" battle.
Also, a trainer could not carry more than six Pokemon. This was for a few reasons: One, as a battling limit. Two, Pokemon carried a small amount of radiation that would become dangerous after six. Three, it is hard to give the proper amount of training and nurturing to too many Pokemon.
Also, in order to nerf Pokemon who could learn many, many moves, every Pokemon was only allowed to "know" four at a time. This also greatly simplified battles.
If you broke these rules or allowed your Pokemon to die in battle, you would have your license revoked for life.
Training without a license was punishable by either severe fines or death. Imprisonment was considered inhumane because being imprisoned in a small space made the prisoners feel like Pokemon instead of dignified humans.
The Leagues then created tax systems throughout the countries, then used that money to open "Pokemon Centers," a place where Pokemon trainers could heal their Pokemon for free using an accelerated rest system, and "Poke Marts," where trainers could use their winnings to purchase Poke Balls and medicines, also mass produced.
The League also funded research labs, where Professors would give children a "starter Pokemon," an endangered but relatively strong species reserved for scientific study. Interestingly, many species evolved to become starter Pokemon, knowing that they could survive that way.
Very recently, Bill of the Kanto region invented the PC system. It converts Pokemon into data and stores them in the Cloud. The League bought up the company and put one in every Pokemon Center.
But why go to the trouble of building all that, one may ask. Every region wanted to manufacture the strongest trainers possible, in the case of war with a neighboring region. They allowed young children to wield incredible monsters and then gave them the technology and infrastructure to do it with relative convenience. The Leagues wanted their region's trainers to be as strong as possible.
Modern Day
Today, the regions have become more civilized and know that they are in an era of peace, progression, and civilization. The most recent war was that between the Kanto and Johto regions that left the populations of both crippled.
Humans find Pokemon so endearing, they consider it impossible that their ancestors could have ever lived in fear of them.
The battling system is done as more of a tradition than a real form of government. As the generations went on and people became more civilized, the position of "Champion" carried less and less political power. It became more of a title than a leadership. The autocracy has been replaced by more of an aristocracy in most regions, where the rich businessmen and scientists rule. Gym Leaders, however, are still valued as community leaders and still carry power.
Globally, the government system is very flawed. There is very little inter-regional liaison. The only inter-regional, non-transportation-related company is the famed and privately owned international police force, which acts against groups, cults, and anarchists looking to seize the unclaimed global power. This has caused conspiracy theorists to call them the tyrants.
Still, they have come a long way since murdering each other for leadership like monsters.