Okay I've only been following the summary on Wikipedia but what else besides the Superman army and the Monitors is used from the Multiverse, from what I can tell it's mostly Darkseid invading and taking over until Batman offs him and then some vampire thing shows up and gets it's ass kicked by the Supermen.
Superman journeys through the multiverse (in
Superman Beyond, which Morrison also wrote, and is included in the trade, since it's important to the story) on a quest to get to the heart of what's going on.
FC has basically two plots: Darkseid's, which is fairly literal. It's "the day evil won", with Morrison bringing out all the standard doomsday tropes to use/analyze them (one of the big things being to make evil look, well, evil and freakish, rather than sexy and cool; see Mary Marvel).
Then there's the more openly metatextual plot, involving the "vampire thing" (a corrupted Monitor), which is the embodiment of gritty/deconstructive influences from the 80s (or, perhaps more appropriately, the more derivative versions that followed). At the end, the Multiverse is, among other things, a representation of infinite storytelling possibility.
The former plot, by itself and played with a more conventional dramatic style (without all the deliberate jarring stuff Morrison does near the end as part of his meta-narrative), is a pretty standard epic event idea (and it has the series' most easily digestible scenes, such as all the stuff with the Green Lanterns, which were my favourite parts of it). The latter is Morrison's weirder side, which tends to separate readers into factions pretty quickly.