As a leftist/liberal who doesn’t know too much about polsci, how would anarchism function on a grand scale? Genuine curiosity.
Anarchism is less a system of functions to be implemented, and more of a governing philosophy on how we build other systems. That philosophy focuses heavily on the expansion of democracy and the elimination of hierarchy wherever possible in order to create the most total freedom in the system. It is not inherently opposed to the concepts of governance or laws as many believe. It usually means focusing on smaller governing units, preferring local governance wherever possible, to give people the most direct control over their own lives. Self-sufficient communities are a major goal here.
The meaning of freedom to an anarchist is wholistic; not just freedom to, but also freedom from. Freedom to pursue your life on your terms, freedom from any obligation or inhibition that would prevent or detract from that goal. This includes, for example, unconditional freedom for all people from starvation, homelessness, or the inability to access medical care. It is an intentionally utopian ideal, that we should strive for something that may not even be possible, because that is how we’ll create the best possible world.
Once upon a time, anarchism was effectively synonymous with libertarianism. That word was bastardized in America to the point that it is unrecognizable now.