Why do pigeons live in cities?

There are two answers to this.

Pigeons that live in the cities descend from the rock pigeon, a bird that has adapted well to domestication. People brought the creatures to the cities for such things as food, pets, and simply for show. Birds that were released or escaped became feral city pigeons.

As to why they normally don't go outside urban areas: cityscapes more closely resemble the steep cliffs from which the rock pigeon originated. It's in their instincts to live somewhere with tall and closed structures where they have been residing for many generations.