As for the first part (how they work), it depends if a zoo is publicly or privately governed. In private zoos (such as Phoenix or San Diego), they basically run everything as far as I understand it. In public zoos run by city (or county) government, they are partners with varying degrees of control. The zoo I volunteer at (Reid Park) is owned by the city of Tucson. The zoological society maintains membership, maintains the website, and funds certain staff members, namely education department and very recently the ticket booth operators. They also run all special after hours events. The keepers and senior staff, however, are city employees. The zoological society has input, but no real control, over the animal collection and new exhibits, which the city staff decide.
The partnership of city/county government and private zoological society has varying degrees of success, depending on the location. The AZA chastised the Los Angeles Zoo years ago for tension between the two groups. And the Rio Grande Zoo maintains two competing websites - one run by the society and one run by the city. Many publicly governed zoos have privatized in recent years (Dallas, Fresno, etc) and it seems to be a positive move when it happens. My zoo may be heading that way eventually, but the process is going slower than some (myself included) would like.
As for the second part of your question (how to start one), you need to form a nonprofit organization recognized by the IRS, which entails having a board of directors. Attracting such a group means you are passionate enough and articulate enough to get the right people to join you. Having good connections and/or money helps too. Of course if you want to start your own private zoo you do not necessarily have to have a zoological society, but it would help if you want to set up the zoo as a non profit entity.