Okay. This is going to be potentially (although I don’t know exactly how much of this is going to come out in the end, it not really being their story) spoilery as hell, so I’m going to use the appropriate tags in places for anyone who doesn’t want to know what’s going on. (Also, there may be some inaccuracies because I don’t have access to my notes right now.)
It’s not a resource-extraction empire of the classic type, because these aren’t conquered provinces. It’s also very important to remember that they didn’t actually intend any of this; they just accidentally ended up there, and when you’re riding the tiger, it’s hard to get off.
There is a certain mixture of extrapolations from Things Not Said But Seen in the inspirational material, and from the problems of the notion of post-scarcity, those being that it’s largely impossible insofar as someone has to do something sometime to make things happen¹ unless you have access to a large number of willing slaves. As far as the extrapolations go - well, for example, you notice all those “agricultural colonies” and “mining colonies” and so forth we see in the outer regions of the Federation? Seems a mite odd for a “post-scarcity” society, 'cause they’re not just the planets of the hobby farmers and people cosplaying grizzled old prospectors.
The start of their problem, way way back in the day - probably, if we steal an historical analogy from their inspiration, sometime in the post-Kirk period - was an outbreak of idealism along the exact lines you might expect from later works: selfless service to bettering humanity, disdain for wealth, benevolent post-scarcity provision for the needs of all, and so forth, coupled with the development of the underlying technologies that would, in theory, make it a lot more possible. The public of the time leapt on the idea of guaranteed necessities and the elimination of poverty for all time, and the politicians followed.
And the first phase of its implementation went relatively smoothly. (Well, except for the nationalizations, but even there, it went through.) The Core (including member homeworlds in general) enjoyed their new guarantees, and the Shell was pacified by the notion that they would be next in this phasing in of the new system.
Of course, there was a problem. More accurately, there were three problems: One, it being more expensive than anyone planned for, and the definition of “necessities” growing on a fairly regular basis (up to the “replicate anything you want” level, ultimately). Two, industries that didn’t care to be nationalized moving operations outward to the Shell. And three, productivity in the Core walking right off a cliff as less scrupulous people realized they didn’t have to work, which meant that more and more of the burden of running the system ended up being placed on the backs of the Shell colonies.
So you can predict the ultimate dynamic this set up. The Shell colonies are kept in the system by the carrot that they will “eventually” get their freebies, and the stick that seceding means losing the protection of the Republic Navy against the pirates and rebels and not!Klingons and not!Romulans and those wicked apotheosians next door who are totally not!Borg. The increasing weight of the remittances demanded of them drives economic stagnation, which causes the productive portion of old Shell colonies to migrate outward into new Shell colonies which start out earning, and said old colonies are then swept into the Core (as part of the “carrot”) and increase the overall burden the system imposes.
(This is a gross oversimplification of a lengthy period of history in which there were ups, downs, attempts at reforms, partial fixes, etc., etc., etc., but this margin only has so much space and, like I said, I don’t have access to my notes right now.)
Even given free expansion, the square-cube law would eventually do this whole mess in, but in practice, given the failure of the Propulsion Group to keep up with stargate manufacture, and that they were running into increasing span-of-control problems anyway… well, they are now in the era where the wheels are starting to come off because, among other things, the Shell no longer has its traditional safety valve (emigration) and the Core War placed far more strain on the system than it was supposed to; there’s been a lot of weakening that no-one at the top noticed until it was upon them.
1. The Republic has this problem aggravated by their prejudices against sapient computers, but really, that’s a sideshow; you’ll note the Empire has a very thriving business community for what amounts to exactly this reason. There isn’t a Culture-analog in the 'verse, but if there were, it would be noted that the Minds work hard and the illusion of post-scarcity is maintained by the willingness of the biosophs to be pets and live only on the terms that their owners permit them.