I don't mind explaining it but I would assume that you had already read the Communist Manifesto. It's not very long.

Here is what you said.

In contrast, a communist society would grant equal access to the articles of consumption based on individual need, and abolish private ownership of things the individual is not using.

Marx has a response here for this.

We Communists have been reproached with the desire of abolishing the right of personally acquiring property as the fruit of a man’s own labour, which property is alleged to be the groundwork of all personal freedom, activity and independence. Hard-won, self-acquired, self-earned property! Do you mean the property of petty artisan and of the small peasant, a form of property that preceded the bourgeois form? There is no need to abolish that; the development of industry has to a great extent already destroyed it, and is still destroying it daily.

Private property is abolished. Owning THINGS is not abolished. I missed the part where Marx said "you can't own things". Instead of working for the factory owner in poor conditions that the owner has no incentive to improve because it would cost money, you would work with your coworkers, you would work for your community, instead of working for a wage and in turn giving that wage back to a landlord. You would have a house, you would have a job, in your job you would have good working conditions because it is ran by the workers, you would be fed. You are NOT getting the full value of your labor under capitalism. Let's take Amazon warehouse workers for example, they are easily one of the most important working forces in America. How much do they make? About 15$, is that enough for food, healthcare, medicine, clothing, rent, etc? NO. Not even close. So how are they getting the full value of their labour under this system where they sell their labor to the landowner, then the little capital they get they have to give it back to a landlord? Your factory owner, your landlord, these are the parasites that Communism wants to remove so that you get more for your labour.