This new Homefront: The Revolution trailer should give you an idea of how the Greater Korean Republic (GKR) got a foothold in the US, and eventually occupying it.
Apex, as you see in the video, started out as a tech firm focusing on computers and smartphones before delving into military tech.
Other factors come into play as to how the GKR was given access to the country, but it became occupiers easily when it took control of the Apex-created devices sold to citizens and the US military. Basically, Apex is in bed with the GKR government.
Homefront: The Revolution is in development at Dambuster, and publisher Deep Silver will release it on May 17 for PC, PlayStation 4, and Xbox One.