There are a few main areas where an HOAs tend to pop up:

• A developer wants to build out a neighborhood on what was previously farmland or whatever and there is no municipality that will build out the roads or utility access, or provide trash services (maybe not even emergency services), so the private developer needs to negotiate the build out with all of the services and infrastructure that make modern living possible, and then arrange for everyone in the neighborhood to pay for those services collectively, from road maintenance to the electricity consumption of the streetlights to the trash services picking up people’s trash.
• A developer wants to build a large building or complex where each unit is individually owned, but that things like the foundations, roofs, walls, stairwells, and elevators need to be commonly owned.
• A developer wants to build out a neighborhood with certain shared amenities available to the whole neighborhood: neighborhood swimming pool, playgrounds and parks, maybe a gate, maybe access to things like a lake or a beach. I’ve even seen one with a plane runway for small propeller aircraft that the homeowners could use.

Lots of these are just a replacement for what a city actually should be providing (infrastructure, parks), or designed to be exclusionary (gated communities, private clubs), or just the reality of shared buildings (typical condo association). A lot of the rules, then, start resembling normal municipal zoning laws or nuisance laws, and a lot of the fees pay for things that look a lot like taxes to pay for community-wide benefits.