It comes down to server tiering and instancing, as others have mentioned. Each shard could consist of a handful of top-level world servers, and beneath each of them could be multiple region servers, and beneath each one of those many instance servers, etc. Thousands of players may be playing your game at once, but each server would only have a handful of players of connected at once (or not have to do much work per player). You'd also probably have separate servers for logging in, chat, auctioning, finding instances, persisting player data, etc., that serve very specific purposes. Basically, the work is spread out across many servers that are each designed to do one task really well.
Of course, the game design itself would have to be amendable to this sort of stratification. If you're trying to design a game where thousands of players can battle it out at once ala EVE, you'll need a whole separate bag of tricks!
By nature, in an MMO there will be a point in which state must be shared between players in some manner. I help design, build and maintain massively multi-user telco services for work on the magnitude of 100k+ users per cluster. Oddly enough this is far less complex than an MMO
As was mentioned, it is always preferable to push out anything not "shared" to a series of load balanced services - i.e. find the smallest thing you can do on a large scale that is not visually time sensitive and do it well, then put it behind load balancers - authentication, chat, UI responses (menus etc.). State always becomes an issue here but many times state isn't as necessary as it seems. Perception is the key concept here and it will be different based on the mechanics of every game but there are some general assumptions that will be made.
Physics and collision detection tend to be things that must be housed in an instance of a specific maximum number of players for example. Proper game design and level/area/zone distribution can do more good than innovative code in these areas many times. Distributing wealth, items, quests across a more vast landscape will allow things to spread out much more than if things are concentrated in certain places. An example of this is "the best x in the game" where "x" is something only found in one spot. If there can be only 1 then the goal of many players will be to acquire this item. If on the other hand, there are many balanced items etc. in the game, players will naturally be divided on which ones they care to quest for the most.
As a service provider, our company has many issues to overcome however uptime/availability and load distribution tend to be the biggest daily engineering feats we manage on a daily basis. I can only imagine that an MMO has similar requirements as they too are ultimately a service provider.