To give an example, lets take a look at the original Star Wars trilogy. The plot line to these movies is fairly simple, there is a group of Rebels fighting against the tyrannical and evil Empire, hoping to free the galaxy from its clutches. However, the story isn't told that way, if it was there would be a focus on the Rebel and the Empire as groups. Instead, that story is told through the stories of the main characters, all woven together and around each other to make the rope that is the whole story. Those threads that make the rope?
Luke Skywalker - The hero's journey. A coming of age story. Luke goes from being the young naive farm kid, to becoming the inheritor of an ancient order of warriors. As the movies progress he comes of age, becomes a man, and faces up to the sins of his father while becoming an icon for the Rebel Alliance. His story is this, starting with him meeting ben on Tattooine, and ending at the resolution of his personal conflict with his father in the Emperor's throne room aboard the death star.
Leia Organa - Leia's primary story is the romance story for the movie. However, she also has the story of the born nobility needing to learn how the other side works. Finding that there are redeeming qualities in those she would otherwise view as scum and becoming grounded and a leader in the alliance. Watch A New Hope and how Leia is pictured aboard the Death Star before the rescue. Then watch Return of the Jedi and how she is with the Ewoks. One of those is a significantly more mature, down to earth woman. That is also the progression of her story, as she finds her place next to Han.
Han Solo - The rogue finding that he has a heart of gold. Han's story is one of a person finding that there are things in the world that are worth fighting for and putting your life on the line for. Friends that are worth going back to help. Han's story is less about maturing and developing as it is about finding out who he really is, and becoming ok with that. Again, look at A New Hope, Han is very mercenary like in that, through Empire he is trying to get off with his pay day but circumstances keep sticking him in trouble. By the end of Return of the Jedi though he is committed to the cause, and volunteering to lead rough missions for the sake of the mission.
There are more threads in the original trilogy, it is part of why the story telling in the movie is so good, but those are the three main threads for the narrative. Individually each of those would be an interesting story, though some things would have to be told differently, but woven together, wrapped around each other, it becomes something more.
Now why is this important for gaming? Well, for one thing it shows you where the focus for your story should be. Most GMs when wanting to run a game have a story that they want to tell. This is all well and good, but the thing is you can't just bludgeon the players with that. If you want to just tell that story, and not have interaction with it, write a book. Don't run a game, write a book. If you want to run it as a game, take that story you want to tell and the main characters, and wrap the main NPC stories around the core of the game, the PC stories.
Your PCs are your games main characters. I used this before, but in a 4 player game there are FIVE creative entities ruling the universe. The four players and the GM. 20% of that team is focused on the entire world. the other 80% if focused in 20% chunks on 4 individuals inside the world. Think about that for a second, of the "five gods" that rule this universe, 80% of the attention is on four specific individuals. If that isn't a description for main character I don't know what is.
Now that we know our main characters, lets look at what our players have given to us to work with. Disadvantages are a great place to start, they show the kind of trouble a player wants to get into. Specializations work too, they show how players want to solve problems primarily. Reading back stories we can get clues, the same for goals and beliefs. Now try to set each player up with their own story. This doesn't have to be a huge thing, but something that is theirs and for them. Give their character a story that could theoretically stand on its own, even if it isn't too strong. Now weave those stories together, wrap them tight around each other. Around this core, wrap the stories of the NPCs and other events in the world.
The players now each have their own things. Their own stories, focus on them with their hopes and dreams, failures and successes. They're involved, and they feel they are doing things because the story has sections that focus on them hunting down their father's killer, or coming to terms with what it means to be a hero. The events of the world and the meta-story (the over arcing story you want to tell) effects these stories, pushing and prodding, bringing up the events of their personal plots. In the end, you don't just have your story, you have multiple stories all woven together into something that is significantly better than the sum of its parts.
The GMs and story crafters that can do this, have experiences and weave tales that have people coming back to them for years to come. In the case of games, it is the game that you'll hear players talking about a few years after it's over. That, by the way, is an amazing feeling, and all it takes to make it happen is to know what makes a good rope.