It explains much if you consider Starfleet as a hobby that only a subset of Earth people partake in, like today we might play paintball or join a bowling league or be in Scouts. So yeah, they don’t use money…in their little “Space Ranger” club.
I like to think they’re the Galactic AudioVisual Club crossed with [del]Sea[/del]Star Scouts, and the rest of Earth society treats them pretty much as such.
This isn’t borne out by the on-screen evidence, you say? “A-V Club!” I reply.
As I’ve argued elsewhere before on the Dope, any organization that has the power of imprisonment (after court-martial conviction) and potential death (when ordered on dangerous missions) over its members, and provides exploratory, diplomatic and military service, as Starfleet does for the Federation, isn’t just a bowling league.
ST in its various incarnations has certainly been inconsistent in dealing with Federation money, but here’s a good list of canon references: Federation credit | Memory Alpha | Fandom
StarFleet is also seen enforcing various Federation laws (as seen when chasing H. F. Mudd). So they have the power of arrest and imprisonment over non starfleet Federation citizens, too.
Originally Kirk was, in addition to everything else a starship captain does, also supposed to be the hand of Federation law in far flung outback colonies and outposts, a sort of combination Federal marshall and circuit judge. But the series was never consistant in how slow/difficult communication with the central Federation was, so it was never clearly established that that sort of plenipotentiary authority was routine.