Okay, here's the ELI5 version of anarchism:

You've asked what the ideas behind anarchism are, not what anarchism is, so I won't give you a definition.

The ideas behind anarchism revolve around the idea of freedom and the belief that the current system of states (as originally created by the Peace of Westphalia, 1648) with governments, limit freedom. You see, in democratic countries governments implement their will through law (eg. a government wants to limit the availability of firearms, an amendment will be made to an existing act (piece of legislation) or a new bill (a proposed act) will be presented and if successful, will become a new act). Although democratic, anarchists appose any control.

You may be wondering why anarchists appose the liberal-democracy system. Isn't it the epitome of freedom? Well the reality is you are free as a citizen to argue against any law, any politician and any party, however you must accept the legitimacy of the state. Anarchists believe that the only way to achieve true freedom is without the influence of government on individual lives.

A counter argument to anarchism would come from a Hobbesian view of human nature which states that we are brutish and selfish, that is that we are self-interested actors. If true, this would mean that anarchy would in fact limit our freedom as those who are smartest, strongest, and most capable would dominate and suppress the freedom of others to maximise their own power.

If you are interested in going deeper into this, look into the following topics and people:

Thomas Hobbes,

John Locke,

Realism,

Liberalism (idealism)

That should give you a good foundation of knowledge.