The motivation for this Sith empire is revenge, I'll describe the reasons for this.
One thousand three hundred years before this game, the Golden Age of the Sith on Korriban had just ended with the death of the great and undefeated Dark Lord Marka Ragnos, with this two Sith Lords emerged as his potential successors, Naga Sadow and Ludo Kressh.
Eventually after a great duel Marka Ragnos' spirit emerged and made Naga Sadow Dark Lord of the Sith, soon after two republic hyperspace explorers appeared and Naga Sadow took advantage and claimed they were spies infront of an impending Republic invasion, so the Sith Empire attacked the Republic and started the Great Hyperspace War.
The war at first went well, Naga Sadow used a Meditation Sphere to spread his skill with Battle Meditation across the galaxy on all fronts, bolstering his own men's morale and depleting the Republic's at the same time, he summoned illusions of massive armies and armadas all across the galaxy, tripling his own forces, until, his rival Ludo Kressh attacked his sphere and distracted him, immediately the war across the galaxy turned back to the Republic's favor, Naga Sadow battled with Ludo Kressh above Korriban, their fleets decimating each other, the Republic arrived and just mopped up the remaining Sith ships.
But the Republic went further, they committed genocide on the Sith People, almost wiping out their entire race and all their planets, but, a few Sith ships escaped, led by the Sith Emperor, they made hyperspace jumps for twenty years before finding Dromund Kaas in the Unknown Regions, and made it their capital of a new Sith Empire, swearing to take revenge on the Republic and the Jedi, for thirteen hundred years they usurped system after system, rebuilding their empire, along with their armies and fleets, then they returned, with the intention of wiping out all Republic civilization.
That is their motivation.
Oh and Darth Bane's rule of two isn't implemented for another thirty six hundred years.
that is one hundred percent right, and in the end it kinda started a cycle of revenge for both sides to be fair