Their early interstellar history is one of mistakes and failures. Eventually they matured out of that stage into a race of interplanetary conquerors with a desire to take down the Galactic Council.
Their attack on Earth was spearheaded by a "fifth column" of spies, the earliest of these was an elderly gentleman known as Mr. Bridwell. They also impersonated officials in powerful government positions, weakening Earth's defenses and creating discourse of trusted institutions and highly respected individuals.
By the time the actual invasion occurred authorities (either infiltrated Enelsians or duped earthlings) had outlawed super-powered heroes, with orders to lock them up, or if they show any signs of resistance, shoot them on sight.
After the Confessor revealed the ruse, they were eventually forced to retreat.
The policing actions of the Galactic Council seems to have neutralized their abilities and powers to invade other planets, but given the vastness of space, and the shape-shifting skills of the Enelsians, one can never too sure.