In order to reconcile the "not using money" bits with the obvious retention of private property and occasional use of "credits" and various other currencies, you need to wrap your head around just how ridiculously wealthy the Federation really is.
In many stores where I live, there will be a "penny jar" on the counter. A penny is worth so little these days that people don't find them to be worth the effort of carrying around and toss them into the jar when they get them as change. Later, someone who is short a few pennies, or who has a bill such that adding an extra penny makes counting the change easier can use them to make up the difference.
Now, take that concept an apply it to a society where a common freighter consumes more power than our entire world does today...
The amount of resources needed to feed, clothe, and shelter a human being for a year is so insignificant by comparison, that you can support an entire planet off the penny jar of the Utopia Planetia shipyards.
Why then do people still work in coffeeshops and restaurants? And then give their product away for free because it's not really worth much? Well, because economics is all about scarce resources. In the absence of modern-day scarcity, it rises to a higher level. The food is practically worthless, just like air is today. But having a reputation as the best chef in the city? That is worth something. Housing in general is practically worthless, but a particular house in a particular place is unique and may, therefore, be valued by a particular person who keeps ownership of it. For the average apartment building though, a reputation as a good landlord is more valuable than the land and the buildings ever could be.
The really scarce things still involve currency. A trip to another world may require payment. Rare and unique artifacts are still bought and sold. Places where wealth is not as abundant still use currency in everyday life. The Ferengi have an obsession with physical goods that cause them to prioritize differently and not really comprehend the Federation system. (Although Quark starts to get the idea in one DS9 episode when he's gone bankrupt with the incredulous line, "My friends are my assets?!")
The translation between reputation and physical currency would be pretty easy. A person within the Federation with a good reputation could easily convince others to part with physical resources (albeit it would likely cost them some of their reputation in the process.) A trader who routinely brings in valuable physical resources will earn a good reputation. "Credits" are probably much like reputation points on this site. Imagine a whole society run that way. It's not that there's really "no money". It's that physical goods just aren't that important anymore.
For a good book that focuses more on the details of a "post scarcity society" like the Federation (but on a smaller scale) I'd heartily recommend "Voyage From Yesteryear," by James P. Hogan.