Yeah that's a long read and the webpage as it is designed itself isn't inviting, @Lettuceeatlettuce@lemmy.ml posted a great comment which might be an easier introduction. I'll just select and copy paste paragraphs from your link that are relevant to understanding anarchism, but I do recommend allocating the time to read the whole thing if you're interested in learning more :

anarchists consider it essential to create a society based on three principles: liberty, equality and solidarity.

Liberty is essential for the full flowering of human intelligence, creativity, and dignity. To be dominated by another is to be denied the chance to think and act for oneself [...] Thus the society that maximises the growth of individuality will necessarily be based on voluntary association, not coercion and authority.

Equality is essential for genuine liberty to exist. There can be no real freedom in a class-stratified, hierarchical society riddled with gross inequalities of power, wealth, and privilege. For in such a society only a few -- those at the top of the hierarchy -- are relatively free, while the rest are semi-slaves. Hence without equality, liberty becomes a mockery -- at best the "freedom" to choose one's master (boss), as under capitalism.

Solidarity means mutual aid: working voluntarily and co-operatively with others who share the same goals and interests. [...] without liberty and equality, society becomes a pyramid of competing classes based on the domination of the lower by the higher strata. In such a society, as we know from our own, it's "dominate or be dominated," "dog eat dog," and "everyone for themselves."

Anarchists do not believe that everyone should be able to "do whatever they like," because some actions invariably involve the denial of the liberty of others.

Anarchists desire a decentralised society, based on free association. [...] Only by a rational decentralisation of power, both structurally and territorially, can individual liberty be fostered and encouraged. [...] anarchists favour organisations which minimise authority, keeping power at the base, in the hands of those who are affected by any decisions reached.

Addiitonally, this is a recommended read : Ruth Kinna - Anarchism: A Beginner's Guide - https://files.libcom.org/files/Anarchism%20-%20A%20Beginners%20Guide%20-%20Kinna,%20Ruth.pdf

Some youtube recommendations : Zoe Baker (@anarchozoe) ; Anark (@Anark) ; Red Planet (@RedPlanetShow) ; AudibleAnarchist (@AudibleAnarchist1)

Just waking up so don't have the brain power to give an in depth answer (Lettuceeatlettuce's reply is god E: good obviously, not god lol.. In anarchism there are no gods no masters!), but one thing jumped out at me:

But watching examples of decapitated states devolving in to warlord rule makes me think the idea does not really work.

The problem with looking at examples of anarchism (or communism for that matter) within a wider capitalist world is that capitalism despises competition and will do anything in its power to destroy it. So capitalist states intervene, either directly by installing a well funded and armed opposition to the anti-capitalists, or they indirectly create war in the region so neighbouring countries can destroy the project, or they impose sanctions making it impossible for the project to survive, and so on... The other option is that the "leader" (which shouldn't exist) can't help but be tempted by the power capitalism can offer (only) those at the top, and they turn on their own project, making it state capitalist themselves, leading to its demise (like the USSR). But that is because we've been socialised under capitalism for so long it's hard to unlearn, not because greed and selfishness are "human nature".

Remove capitalism entirely, and re-educate people with our natural instincts of cooperation and community, and things would turn out very differently..

Counter measures against warlordism would be crucial for effective anarchism for sure!

I think the easiest approach to anarchism is searching for Chomsky talking about it.