In the context of a city, self-rule means the place has a mayor and a city council. It can pass local ordinances, have a local sales tax, and provide services to the constituents.

Not all places start out with those things. The rules differ by state, but generally, you need a certain population, a percentage of whom must petition the higher authorities (which you mention), and then vote as a community to create the create the infrastructure and take care of themselves.

At that point, they lose a lot of the benefits provided by those higher levels, like the county and state service providers of fire, police, and garbage disposal for example.