Even when looking into the real world, we can see small communities that form within larger ones. The regulars of a bar, the cliques in schools, and of course, the personal networks that people have. What I mean is, someone may well be standing in the middle of a group of people that they are unfamiliar with, talking on the phone with someone that they do know. For all intents and purposes, those other people around him have not entered into his personal network, and thus they have not entered into his community. This really is similar to what you see in social networking sites: there are a lot of Facebook and MySspace users, but your own community on those sites are much smaller. Well, this isn't the greatest example, as there are people who will often accept someone as a 'friend' without ever sharing any communication, of course. Still.
What does this have to do with dividing up the community in an MMO though? If people like to live in smaller communities, then the game is effectively providing them that. When players are divided into factions, they have a narrower group of people that they are playing on the same side with. While this may sound negative, it also provides more of that 'small town' and 'tight knit' feeling that can strengthen that community. You are also more likely to have commonalities with the other people who picked the same faction that you did.
So, why not just make server populations smaller? Well, why doesn't everyone just live in the country side? Of course, that isn't practical for everyone, but there is something to be said for being involved in something that a huge number of people are a part of. While your in-game faction may only consist of a third of the game's population, you are still involved in something that even greater numbers of people are a part of. You may not bond with your enemies (or bond as well with them), but you do interact with them and take something out of that experience.
Speaking of being involved in something larger, that points out another reason why player factions can increase the strength of a community - perhaps far greater than just the smaller community does. This is the feeling of working together with your faction.
Much like sports rivalries, school rivalries, and even rivalries between companies can help bond those on the same side, so to do rivalries within a game. It is not just the simple fact that a rivalry is present and that you must defeat your enemies, rather it is the fact that you are now part of something larger. You are on a side, with numerous other people, who are working toward a goal. That goal may well be to win a PvP battle against another faction, or it could be another goal entirely. Getting your player-base focused on working together to achieve something builds a great deal of pride within them, bonding them closer and strengthening the community greatly as a whole.
Of course, having a goal to work toward in a collaborative manner does not necessarily imply factions of players. After all, the player-base could be fighting against a single NPC force, or some other set up. That is true. Players could be fighting as one whole against a single NPC enemy... but who is their rival? Yes, in a way, the NPCs are, but at the same time, killing (or whatever) the NPCs is what must be done to achieve the goal. While that alone can do a lot to strengthen a community, pushing it further and giving the players another group of players to work against can do a lot to build up greater excitement, greater involvement, and tighter bonds.
This can be done with direct PvP, but this can also be done with a PvE model as well, such as Final Fantasy XI's system. One way or another, however, player factions can do a lot to help a community grow strong, as strange as that may seem.