Long long ago there was a hyper-advanced civilization that had colonized the galaxy. They are called the Precursors. Eventually, they split into two races who went to war with one another. Just as the final battle between the two was being fought, they disappeared. This was hinted at in the first game but not truly explored.
Thousands of years later, new civilizations slowly started making their way into space. But they lacked the ability to travel great distances. They built a series of "star gates" that enabled ships to travel between them but it was a massively expensive and inefficient mechanism and didn't really enable colonization as much as it allowed for different races to have some basic relations.
The humans, however, were the youngest of these new races and they also were much more reckless in terms of trying out new technology. They adapted the star gate technology to design ships with drive systems that effectively could travel much faster than the star gates could and weren't reliant on them. In short, ships could travel anywhere in the galaxy on their own.
However, this has a negative side effect - now all the civilizations were in a race to colonize and claim as much of the galaxy as fast as they could.
The story of Galactic Civilizations I dealt with that race to claim the galaxy.
In Galactic Civilizations II, we now have the consequences -- the vile and evil Drengin Empire have emerged as the dominant civilization. And they are beginning to conquer the other civilizations. In The Campaign game (which is different from the regular sand box mode) the player has to go and try to unite the other civilizations to fight off the Drengin. But in the midsts of this, they will all discover that there are far worse things in the universe than each other.