I think there are several factors when we ask ourselves why there are so much fans of this series (including me, heehee). One has to consider the fact that Squaresoft made these games for the Japanese audience. However, when translated into english, a huge uproar among US fans began for these games especially after Final Fantasy 3 (6 in Japan) for the Super Nintendo Console. This put Squaresoft on the world map, and they had to pay a bit more attention to their world-wide audience now that they were producing for the playstationm beginning with Final Fantasy 7.
One of the main reasons why the FF series are great RPGs is because they have good storylines. They got out of that wacky "the princess has been kidnapped, go rescue her!" plot that was getting a bit overused in Zelda, Dragon Warrior, etc. and instead brought new concepts such as multiple character partys, huge variety of equipment, character classes (black or white mage, warrior, thief, ninja, caller, etc.) or as in Final Fantasy 6 a complete custom-based style that allowed us players to create our own strengths and weaknesses for each characters with espers.
Another factor that is underrated but I think that is very important is the music. I know it's only VG music, but it is simply awesome and keeps the player in the story. The number of great themes composed in this game can also explain the longevity of the series since it has only upgraded in soundquality, and the number of great themes did not decrease either.
About point three I must make a remark: there is also two music themes that make an appearance in each final fantasies, which are the prologue and the prelude. Aside from that though, there is not much connection between the final fantasies except that the party you control has to overcome a fierce battle at the end with a guy who wants to control/destroy the world, and you are always aided by a guy named Cid.
There are alot of people who think that Final Fantasy series are the "peak" in RPG gaming, although there are numbers of good games out there that are less commercialized. From #1 to #10, they all have/had appealing graphics, incredible storylines (except the first ones....... corny!), breath-taking music, and the challenges imposed by the game were not too hard but not too easy, although there always have been sidequests which required great amount of time to level up and such. This aspect pleased every type of audience, from the rookie ones who were interested by the storyline, and the "hard-core" players which made competitions of determining who would get to 9999 Health Power with every character...
That was my 2 cents canadian on this...