Otherworld Invaders are a core part of gauntlet mode. They are a group of 5 stronger than average officers that use the same formation abilities that you can use. Defeating them tends to give you rewards in the form of rare orbs or ingredients in making that officer's ultimate weapon.
They appear in one of two ways:
1) Staying on a battlefield long enough
2) Opening a treasure chest that has been altered by miasma (marked Red on your minimap).
The higher a battlefield's miasma level, the faster the rate both of these happen, with an even larger rate for playing a battlefield shrouded in miasma (Red text in level select).
There can be only one group of Invaders on the battlefield at a time. Opening a chest will teleport the current group to the chest's location.
Simply encounter a group of these and destroy them and the achievement will be yours.