Amenities can come from a wide variety of sources. Each type of Luxury resource can contribute up to 4 Amenities to your empire (1 per city) to the 4 cities that need them most. Distribution happens automatically, as long as the Luxury resource is improved (a Plantation improvement built on Spices, for example). The Arena, built in the Entertainment Complex, provides Amenities to its local city. The Zoo and Stadium also provide Amenities, but regionally. Some wonders also provide Amenities. Furthermore, there are a few Religious Beliefs and policies that provide Amenities. Some Great People also provide Amenities.
War weariness and bankruptcy will lower the Amenities in a city.