In STO they call themselves the Terran Federation, not Empire.
I think new weekly series with them should answer these questions, or maybe a proper story arc in Season four. They kind of just show up with little explanation other than that they tried to invade our space with the help of the True Way and that presumably there is a power vacuum left from the removal of O'Brian by his son and then the removal of the son by us.
There is a VERY old comic that gives a huge amount of info on this subject. I think it was about 100 pages or something big... I'll look up the name later...I don't know which comic bin it's in :p