People of inexplicably confident certainty that they alone have discovered legal loopholes enabling them to ignore laws they don’t like while benefitting from public services and infrastructure of a nation they assert they are not a citizen of. It’s delusional ignorance.

The core belief of sovereign citizens – initially, anyway – is the notion that since government exists at the consent of the governed they can “opt out” of being subject to the laws of wherever they are.

This has a tiny grain of legitimate logic to it, in that not a single person on Earth is given a choice of society and/or country to be born into. Governments attempt to exert absolute authority over everyone within their spheres of influence regardless of what those people may happen to think of the matter, and the feasibility of them physically leaving said society/country notwithstanding. All laws are just words on paper, after all, and from a certain perspective completely artificial, arbitrary, and transient.

Where it all breaks down is that these people typically arrive at the above conclusion by being absolutely stark raving loony, and typically want to have their cake and eat it, too – they don’t want to be subject to obeying laws, or paying taxes, or having to register their vehicles and get driver’s licenses, pay child support, etc., but they still somehow feel entitled to the use of public infrastructure like roads and bridges, police and fire services, municipal water and sewer use, and so forth. In modern times a simple “no gubmint can tell me what to do and I’m answerable only to myself” outlook has mutated into this arcane and nebulous pseudo-religious willful misinterpretation on the wording of laws, what is and is not printed CAPITALIZED on various government documents, and fixation on “contract law,” treating every interaction between everyone and every thing as a “transaction” which the sovereign citizen believes is inherently negotiable (always in their favor, of course).

This is furthered by shucksters who sell books and seminars to idiots the types of people who have the right type of chip on their shoulders, which purportedly contain the secret knowledge and legal incantations to make all this work but are, of course, just bullshit. Usually people who entangle themselves in SovCit bullshit are trying to weasel out of of some particular financial obligation. Not wanting to pay child support seems to be a very popular one, as are taxes in general, fines, loans, and liens.

The whole thing is just fascinatingly whacked the more you look into it. Here’s the RationalWiki article on it, for instance:

https://rationalwiki.org/wiki/Sovereign_citizen

Further to the other answers, “sovereign citizens” are an interesting variant on a “cargo cult” mindset.

The cargo cults, if you’ve not heard of them, came about after WWII. The Allied forces, advancing through the Pacific, set up airbases on various islands. These islands had tribes living on them. The tribes got a crash course in the wonders of modern society, American army style. Unfortunately, the gap between their experiences, and the world they were now exposed to was huge, and brief. A lot of misunderstandings were made (either due to insufficient background knowledge, or bored/malicious information from the troops involved).

When the allies upped stakes and left, the tribes were left a little shell shocked. They had the bright idea that if they recreated what the Americans had done, they too could summon the metal birds from the sky, full of a vast wealth of cargo! They then went about reproducing everything they had seen. They built runways, control towers, and fake planes, to bait down the cargo planes. But it never worked! They obviously weren’t doing it exactly right. So they tried harder, recreating all they could as closely as they could.

Now to us, this seems crazy. Of course you can’t summon a cargo plane by sitting in a wooden “control tower” talking into a coconut! We have a larger context however. We know that those planes were sent, not summoned etc.

“Sovereign Citizens” have a lot in common with these cults. However, they are focused on the legal system. Most legal systems are convoluted and arcane. They are less designed, than accreted over time. Lawyers, and the hyper rich who imply them, use this to run rings around the systems in place. They used complex legal entities to game the system to their advantage.

“Sovereign Citizens” see this and thought “why can’t we do that?”. Unfortunately, they didn’t understand what was actually happening. They tried to recreate it, but lacked fundamental information. Even worse, a number of grifters found them, and decided they were excellent marks. They fed them additional bullshit, and gave them ever more complex instructions to make their plans work. When they failed, it’s obviously because they did it wrong, or got out-spelled, not because the instructions were BS to begin with.

In short, “Sovereign Citizens” are a mix of the desperate, the stupid (not always the same thing!), the brainwashed and the grifters, all wrapped up in an almost religious cargo cult.

That phrasing refers to a very broad set of movements and individuals. The usual core beliefs are:

• Legislation in their jurisdiction and the government’s authority to enforce it is in some way defective.
• People in their jurisdiction can opt out of laws and government, and live only under “natural law”.
• People have to perform a set of legal procedures (spells, effectively) in order to achieve that.

Exactly why and how law/government authority is defective, how they understand natural law, what the spells are that they have to cast - all of these are extremely variable both between jurisdictions and between individuals.
Primarily it’s a set of grifters charging money for courses and materials to learn about these beliefs from whoever they can convince. Sometimes, as in Germany, it’s a group of neo-Nazis plotting to reinstate the Kaiser.

You might enjoy münecat’s longer form explanation.

Basically a bunch of people who claim to not be citizens of any nation and will therefore try to claim they essentially have diplomatic immunity no matter where they go. Combine that with the personality and knowledge base of a meth fueled jailhouse lawyer. Actually, that’s an insult to meth fueled jailhouse lawyers and I take that back…