Fallen Empires is a NPC similar with Pirates. While Pirates generates stations so you can make some profit, Fallen Empires has special planets or megastructures that you can conquer and be integrated into your empire.
Fallen Empires can generate planets (Machine and Gaia biome), Ringworlds and Habitats. Each player have their own special planet/structure to conquer and coordinates are not transferable between players, since player can only observe in the galaxy the planet/mega dedicated for him.
If you spy them, you may notice something called "Prosperity" or "Unrest". While this value is closed to 100%, it means the planet is loyal to Fallen Empire. If it gets lower than 15%, on your next attack, the planet/mega will become yours. To reduce this value, you need to select "Conquer" mission to attack Fallen Empire. At end of each fight, you will receive a message of value drop.
If you conquer a planet/mega, Fallen Empires will attack you back to reconquer the planet you took from them. Expect heavy artillery.