The Garleans were pushed from southern to northern Ilsabard ~800-900 years ago. Some of their ancestors may have come from south-central Othard, where the Ivalice of FFXIV was located.

The reason for the Empire's expansion beyond Ilsabard (which they hold some ancestral claim to; whether or not it's enough to justify their actions is open to interpretation) was stated to be preventing the summoning and spread of eikons due to the dangers they pose. However, this is simply a lie that was sold to the common people by Solus and doesn't hold up under scrutiny - the only real reason for the Empire's continued aggression against every other nation is to cause Calamities. (The previous and currently reigning Emperor have different motives for doing so, but the means to their ends are exactly the same.)