SC lore and plot has undergone numerous revisions and retcons. For example, the earliest previews such as the first teaser trailer on the WC2 CD and the website at the time explain that the terrans are a collapsed empire who are engaged in a cold war with the protoss until the zerg invade. In the manual, Koprulu is a lost prison colony composed of the survivors of a nazi takeover of Earth. According to the original BW website, the Dominion was going to enter a civil war as a conspiracy worked to overthrow Mengsk. In the BW game/manual, we are instead introduced to the UED, a new non-nazi government of Earth that was created by a retcon. Etc.
To recap, what follows are my thoughts on how the SC story should have gone (including a recap of the backstory that wasn't included in the game):
In the beginning, there was an ancient civilization known as the xel'naga. They ruled the milky way for millions of years and experimented on its inhabitants as part of a plan to create the ultimate lifeform. The first successful experiment was the protoss, natives of the planet Aiur in the galactic rim, who exhibited a quality named "purity of form." This meant they had great psionic potential, enough that they might evolve beyond the constraints of physicality. These protoss had an instinctive telepathy that let them exchange thoughts and feelings, the ultimate empathy. This let them rapidly build a global civilization out of what were originally scattered tribes. The xel'naga felt the time was right and revealed themselves to their charges, whereupon they were worshiped as gods and in turn tutored the protoss. With this help, the protoss build a glorious galactic civilization united by their empathy network, the Khala. In time, they grew decadent and fell prey to misplaced nationalism. The tribes withdrew from the khala and turned their gazes inward. The xel'naga were disappointed, labeled the protoss a failure, and prepared to leave for a new experiment. The protoss got wind of this and attacked the xel'naga. While many xel'naga died to their former worshipers, most survived and fled into the void of space. Shocked, the protoss tribes turned on one another and fought a vicious galactic war that came to be named the Aeon of Strife. Ultimately, they were reduced to stone age primitives and space pirates.
Meanwhile, the xel'naga traveled to the ash world Zerus in the galactic core. There they experimented on primitive telepathic insectoids named zerg, characterized by "purity of essence," a drive to evolve. Originally tiny insignificant burrowing worms, after the xel'naga tampered with their evolution they grew larger and stronger. Unprompted, they became parasites that took over the larger organisms of their planet, and over generations genetically modified their hosts into vicious killing machines. Worried that the zerg would be ruined by individual egos like the protoss, the xel'naga modified their telepathy so that the zerg thought as one and furthermore programmed this single ego to obey the xel'naga. This gestalt consciousness became the Overmind, and it created subsidiary intelligences to carry out its will more efficiently. Eventually the zerg consumed all life on Zerus, and turned their attention to space. They transmitted a telepathic message to lure new species to Zerus in order to leave. This attracted some nearby space whales, who were assimilated. Now the zerg could expand into orbit and eventually the galaxy.
The xel'naga were overjoyed with the success of the zerg. Until the zerg suddenly attacked them. The zerg no longer obeyed, had evolved beyond the control of their creators. The zerg slaughtered the xel'naga and consumed their knowledge of the galaxy and genetic sciences. The zerg swarms spread out across the galaxy, consuming everything in their path. Only one thing prevented them from achieving total domination: the protoss.
While the xel'naga had been busy experimenting on the zerg and then gotten themselves eaten, the protoss eventually came out of the aeon of strife. A prophet named Khas rediscovered the Khala and built a religion around it to protect the protoss from themselves and to master their abilities. The protoss rebuilt their interstellar empire and rejoined the disparate tribes. They made alliances with other spacefaring species, secretly watched over less advanced civilizations, and dealt with xenomorphic threats to galactic peace. But some tribes refused to join the Khala and after bloody conflict were exiled into the darkest depths of deep space, where they were forced to develop new sciences to protect themselves. While still nowhere near their height under xel'naga tutelage, they were the most advanced civilization in the galaxy and were more than capable of destroying the zerg.
The zerg watched the protoss for centuries with living probes and telescopes. They knew exactly where the protoss homeworld was located, but they couldn't risk attacking in their weaker state. So they scoured the galaxy for another species with uncultivated purity of form that they could take for themselves and use to destroy the protoss. When they were close to despairing of ever finding such a species, they finally did: humanity.
Centered on their homeworld of Earth, humanity was a warlike species that had spread to countless worlds. They were destructive and left ecologically devastated worlds in their wake as they sought to feed their endless hunger for resources. The protoss knew of them, but considered them unworthy of interactions at the present time; though that could change as the terrans expanded into protoss space. The zerg sent scouting broods into terran space to study them and prepare them for assimilation. The terrans were a fairy advanced species, but nowhere near the protoss, so the zerg believed they would present little threat and could be easily consumed before moving on to the unsuspecting protoss empire.
Unfortunately, even the best plans fall apart. The terrans discovered the zerg and, depending on the proclivities of colony in question, attacked them or experimented on them. The zerg were forced to attack before they were ready, throwing the terran worlds into discord and chaos. Things grew more complicated when the protoss discovered the zerg presence and sent fleets to scour the infestation. Numerous terran colonies and their inhabitants were incinerated, provoking the terran governments to launch a counterattack against the protoss fleets and counterinvasions of protoss space. Some protoss were horrified by the loss of terran lives and tried to stop them fellows from arbitrarily exterminating humanity. It was a clusterfuck.
It's impossible to convey the full extent of the conflict without writing millions of books on it, so I'll limit the focus here to the events in just one sector: Koprulu. It is not a unique or special sector, but one of many caught in the conflict. Events here don't decide the fate of the universe.
Koprulu is a war-torn sector in which the local polities the Confederacy, the Umojans, and the Kel-Morians fought for domination. After the KMC lost the Guild Wars and was absorbed into the Confederacy, the Umojan Protectorate was forced to act covertly by secretly funding the Sons of Korhal resistance movement. Then things changed when the zerg and protoss attacked. To Koprulu's complete surprise, a UED warfleet led by Admiral Gerard DuGalle arrived and instituted martial law. When the Confederacy leaders on Tarsonis inexplicably refused to help the UED deal with the aliens, DuGalle had them all summarily executed for treason and took control of the Confederacy.
Although the Kel-Morians and Umojans were ecstatic and threw parades, the SoK were angry and bitter that the UED didn't step in over a decade earlier when Korhal was nuked for revolting. So their leader Arcturus Mengsk secretly worked to destroy the UED efforts by using zerg terrorism. According to secret Confederate research he had stolen, the zerg would be attracted by psi-emitters over interstellar distances. So Arcturus Mengsk set up psi-emitters in various strategic locations and set them off. To his shock, the zerg took no notice of the emitters, almost as though they knew these were decoys. These beacons did, however, attract protoss scouts. Mengsk found himself being hailed by a mysterious protoss individual identifying itself as Ulrezaj...
I'll stop there. There's a bazillion other ways to do it. Truthfully, I'm not interested in the canon characters and would prefer to expand the universe a lot more.