This is called mutual aid, and is one of the fundamental underpinnings to anarchist theory. People will help each other, generally, if doing so is not too much of a burden on themselves. Systematizing that behavior into genuine community support for one another would (hypothetically) lead to outcomes of increased wellbeing for all.
In your example, the person who left out the chairs isn't worried about being paid back for the chairs. Someone has excess, they shared it freely without expecting anything in return, and the community is better for it.
Couldn't we do slightly better with communication though? Instead of altruistically clearing the dishwasher in the office every time, we can have a rotating dishwasher duty. Is this opposed to anarchism? What about not having an explicit dishwasher duty, but just a socially-enforced expectation to contribute?
When it's a duty that others have imposed on you, it becomes a form of oppression rather than free action. Instead of feeling good about helping everyone, you feel bad about being bossed around by everyone.
The "socially-enforced expectation" is the same. You know whether you're doing something altruistically for the common good, or whether you're doing it because if you don't others will try to shame you.
Sort of, anarchy does not mean individualism. It absolutely can be built around syndicalism and federation. If a group of people says, "we believe it's best to take turns doing this labor" then they can do that.
Typically there's a cultural expectation of "wellbeing for all", which becomes a significant social expectation. If it's the cultural standard, it changes the relationship to chores significantly - I can contribute to them knowing that the general person out there is also doing chores with my wellbeing in mind.
I recognize I'm completely ignorant about this topic so I have to ask: how does the theory fit within the story in the post?
If I understand you correctly, mutual aid theory explains the author buying cheap forks just to do something good. At the same time, in the absence of penalties people will steal forks until barely any remains. So why should one believe that the balance will favor the fork buyers more than the fork stealers?
The basic theory is that when something is abundant and everyone has what they need, the incentive for theft doesn't really exist. If you set out a barrel of forks and say, "take what you need" then at some point everyone has taken as many forks as they need. If we assume this is also happening for other bulk items, and everyone can just have as much as they need, there's nothing to trade those forks for, so taking a bunch doesn't benefit you at all.
Ask yourself, why would you take more than you need? Not some hypothetical other person, but you, personally. Do you think other people are in general likely to behave like you? (Put another way, do you think you are fairly typical when presented with a "take what's fair" scenario?)
It's like the candy bowl at an office or someone's house, or the "take a penny" cup at a store. I'm sure occasionally someone takes everything from it, but by and large people take what they'll personally use.
In your example, the person who left out the chairs isn't worried about being paid back for the chairs. Someone has excess, they shared it freely without expecting anything in return, and the community is better for it.