MMOs are a bit of a special case in how the game tells a story. Most MMOs are also RPGs, so I won't retread that ground. But one area where MMOs excel is telling the story of a world, rather than a handful of characters.
MMOs tend to have absolutely massive worlds to explore, with hundreds or thousands of non-player characters to talk to and hundreds of locations to visit. With all the effort that goes into creating this kind of world, you end up with enough lore to fill several books.
Of course, most people wouldn't actually want to read those books. Most people just don't have the interest or the attention span to read an entire history of a world, with very little to relate to in terms of personal stories. It's the same reason why most people wouldn't sit down and read a history of their own country.
The trick that MMOs pull off is to hide tidbits of lore all around the world and encourage you to explore all the nooks and crannies. Rewarding exploration makes it fun to find the lore, such that players who might ignore a lore database will excitedly try to find the location of an ancient library and search it for tomes of knowledge.
A world's history isn't strictly linear. You can't really say "this happened, and then this happened, and then this happened" without leaving out entire continents worth of history, since so much is happening all at the same time. And so, by hiding bits of lore around the world, MMOs don't restrict themselves to a linear mode of storytelling.
In a sense these games mimic real-life archaeology. There's no central database you go to to find everything - at least, not initially. To create that database you have to go out into the world and discover history, and this is what you do in an MMO. Clever.
The best is when that lore has major implications for not just the past, but the future. As an example, I was playing Guild Wars 2 recently, and spoke to a scholar at the Durmand Priory headquarters. She told me that rather than the four elder dragons we currently know, this ancient text mentions SIX. What?! If we only know about four now, what happened to the other two? Are they far away? Are they still asleep? Were they defeated during the last cycle of awakening? WHAT'S GOING ON
Another major strength of MMOs in terms of world history is that they can change over time. World of Warcraft's Cataclysm brought major changes across the map, so that players who joined before the Cataclysm are now a part of the world history, remembering things that no longer exist in their old forms. The original Guild Wars took advantage of its instance-based world to add a quest line that has you purging infected creatures from the world and actually having them stay gone.
Of course, just like RPGs, the flaw with storytelling in a big open world made to explore is that there's no urgency. But that same lack of urgency is a boost to the story of the world: it lets you explore and discover at your own pace, only rushing if you want to.