The Dual Power strategy seeks to overthrow the current government, not by fighting them, but by out performing them, providing services the government is supposed to provide in a better fashion, or providing services for people that want them, that the government has forbidden.
Sometimes this strategy is adopted out of necessity. If the government is unwilling or unable to help a group of people, quite often those people will help themselves, even if it means breaking the laws of that government to do so. When that group of adventurers agreed to help that town with their bandit problem, because the corrupt local sheriff didn’t think the town deserved protection? Dual power in action. Later on, when those same adventurers escorted that cleric who ventured into the beggar’s quarter of the city to treat the plague, even though the city government had quarantined that part of the city? Dual power again.
Other times, this strategy is adopted less as a conscious decision to wrest power from the State, and more as a decision to fill a need, and profit while doing so. Smugglers, black marketeers, and bootleggers are all engaging in commerce without the permission of the government. Have your characters always visited customs when entering another kingdom? Do they always report every copper piece they loot from a dungeon treasure hoard to the proper tax authorities? While Anarchists can and do engage in this sort of activity as part of a conscious dual power strategy, quite a few who do have no strategy or philosophy other than to make as much gold as possible.
The theory behind this strategy is that, at its core, governments present themselves as service providers for their citizens. The promotes the idea that they are the only ones who can provide the services, and that if they don’t provide the service, or prevent you from getting the service, that’s for your own good. The Dual Power strategy shows that idea to be a lie. When people learn that it is a lie (so the theory goes), that they don’t need the government to get what they want or need, and in fact can get it more easily without the government than with it, then support for the government will fall, and with it the government itself.