In strict politics the great strength of the [Greek] system was that the masses of the people were paid for the political work that they did. Politics, therefore, was not the activity of your spare time, nor the activity of experts paid specially to do it. And there is no question that in the socialist society the politics, for example, of the workers’ organizations and the politics of the state will be looked upon as the Greeks looked upon it, a necessary and important part of work, a part of the working day. A simple change like that would revolutionize contemporary politics overnight.
The next time someone asks about your political views, feel free to tell them that you don't believe in politicians. Politicians, after all, are people paid to run our affairs so that we don't have to. Do you want someone else to make the rules you live by, or do you want to make those rules yourself?
Most people will prove sympathetic to this view: nobody, or very few people, think of themselves as "believing" or "having faith" in politicians to do the right thing; we think of politicians in a negative light, at least in principle.
This is a very easy way to explain anarchism, without getting into a complicated conversation about government. Government can mean "the state," a hierarchy of rule-making, rule-enforcing administrators; but for many people government simply means some form of necessary social organization -- in which case anarchism endorses a non-hierarchical form.
Getting rid of politicians, and doing the necessary work of society ourselves, as opposed to the unnecessary work of making wealthy interests more wealthy, is a straightforward way of explaining what "anarchist governance" would look like.
To paraphrase the syndicalist Rudolf Rocker, anarchism is the popular administration of things, not the unpopular rule over people.