Oh, you're in for a treat.
They are the centerpiece of a broad category of nonsense. Think snake oil, only applying to English-derived legal systems (they seem to be most prevalent in the U.S., Canada, and the UK). Broadly speaking, they believe based on various logical leaps and historical inaccuracy that the government doesn't actually have any authority to them. Where it often comes up is in the case of income taxes and traffic stops.
Basically, they have various theories about how they're not really subjects of the federal government. Sometimes it's that they're only citizens of their state. Sometimes it just applies to courts (which is where they usually end up); for example, the idea that federal courts can only sit in admiralty (i.e. maritime law), and so have no jurisdiction outside of that. Usually it's more about how these courts don't have any jurisdiction over them. The IRS is a frequent target.
There are a lot of ways they go about this. Some try to selectively renounce their citizenship, for example. They're also big believers in the idea that if you write something a certain way in a court document, this has different effects (whether it's a different color, using odd punctuation, whatever). They also like to cite to the Uniform Commercial Code a lot (odd since the UCC itself has no legal effect whatsoever).
Usually they're just cooks, but every once in awhile they can be dangerous. There have been instances of these folks attacking police or using violence on others, and of course the most famous example is the Oklahoma City Bombing, where a couple of anti-government types set off a car bomb at a U.S. government office building, killing 168.
Any reasonable-sounding base for that one?
None as far as I can tell. It's something I've seen on TV a lot, but I think that's a response to the trope rather than the cause of it.