In thinking about the change -- showing many more ships in DSN than in TNG -- here's a simple explanation, taken from history, to a degree. Before the United States entered World War II, it wasn't quite ready to engage the Axis powers. Franklin Roosevelt put the U.S. on a war footing before 1941, building ships and helping the British. Then, after Pearl Harbor, the US, now in a war, ratcheted up its effort, with many men volunteering for active duty.
I maintain that because of the Borg threat and the events of Wolf 359, Starfleet was already making plans for a Borg incursion -- witness the Defiant
. After the events of The Search
, and with Picard's reports on the Borg and Sisko's reports on the Dominion and the Founders on their desks/terminals, Starfleet's brass, and Federation civilian authorities, decided, "We better start improving our fleet strength to deal with these threats." We have no idea if there was any increase in enlistment in Starfleet, too, although that would have been interesting if they explored that.
So it's simple: Increased threat equals increased size of Starfleet's forces.