Broadly, it's the same as any other faction. Do work for them. Do not do work for the other factions in the system. There
are a few little quirks and foibles that set them aside though. For a start, you can forget about bountyhunting, and not only that, you'll likely have bounty hunters passively working
against you, intentionally or not - it's a good moneymaker in the early game, and later on, if there are players operating in the neighbouring systems, you'll probably have them coming in and doing pirate lord assassinations against your faction even if you don't have any res sites. Most of the players that incidentally/accidentally hurt anarchies aren't doing it deliberately, it's purely down to the fact that legal careers and bounty hunting pay more in this game than crime does.
You can murder 'em (or talk to them if you're feeling diplomatic) if you see 'em, but most of the time you won't even see anyone who's opposing you thanks to instancing, coincidence, timezones, modes, and so on, and PvP is one of the least effective means of working the BGS in terms of numbers, its value is really only psychological in terms of making people think it's not safe to fly in your system in open.
If you don't mind getting your hands dirty, you can harm the other factions in the system through good ol' murder. A compromised nav beacon or haz-res site are good for this if you don't want to have to deal with cops and ATR, but be aware you'll need a safe port to dock at, either in a neighbouring system or a port in-system that isn't controlled by the main faction. Alternatively, if you feel like going nuts, take a murderboat and blow up cops. Be prepared to high-wake when ATR show up though.
Once you get control of a single port, economic activities like trading and exploration data will help an anarchy just like they would any other faction. Black market trading also boosts influence for anarchy factions in their controlled ports (unlike any other faction, where it harms them - so pirating stuff and selling it to an anarchy controlled port will help them, while smuggling illegal goods into the non-anarchy stations will hurt the controlling faction)
@Phisto Sobanii is the go-to guy around these parts for asking about anarchies though, him and his crew have been shaking things up over in Colonia.