After Robb declares himself King of the North, Bran becomes a prince by default, and they rule in Winterfell in his absence. You'd think this would make any boy thrilled—after all, he's the boss—but Bran has to manage the affairs of Winterfell, care for his wild, younger brother, Rickon, and deal with the fact that he has been crippled. That's an awful lot for a kid of ten to take on, especially with his father recently dead and his mother far away from home.
If you recall Bran's story from A Game of Thrones, then you'll remember that Jaime threw the boy from a tower, injuring Bran and causing him to lose the use of his legs. This is a huge life changer for anyone to deal with, and Bran is still learning to do just that. Although he knows he will never be able to walk again, Bran struggles with two competing mindsets—fantasy and depression—throughout the novel.
He fantasizes what he could do if he could walk. While watching the Freys and Rickon play "King of the Crossroads," Bran sits sullenly and thinks to himself, "If I had my legs, I'd knock all of them into the water. […] No one would ever be lord of the crossing but me" (5.Bran.66). But during his audiences with the Winterfell vassals, Bran thinks how he never wanted to be a prince but a knight. Wondering why he has to sit through the meetings, he reminds himself that he is broken and could never be a knight (17.Bran.4). Sigh.
Bran struggles with two potential futures: In one, he is to stay and be the Prince of Winterfell; in the other, he is to literally follow his dreams and discover his latent powers as a warg.
In the first, Bran is a symbol of Winterfell because he is a member of the Stark family. During the harvest festival, Bran rides Dancer to the cheers of the crowd, but Bran knows "it [is] not truly him they shouted for—it [is] the harvest they cheered, it was Robb and his victories, it was his lord father and his grandfather and all the Starks going back eight thousand years" (22.Bran.3). Here, Bran's duty and identity is to his family and his lineage.
In the second, Bran will follow Jojen and Meera and discover the truth of his mysterious dreams and the enigmatic three-eyed crow. As Jojen tells him, "A knight is what you want. A warg is what you are. You can't change that, Bran, you can't deny it or push it away. You are the winged wolf, but you will never fly" (36.Bran.52). This path, then, is all about Bran himself.
It is between these two duties—the duties to family and the duty to self—that Bran struggles throughout the novel, as well as with his paralysis. After the destruction of Winterfell at Ramsay Snow's hands, Bran's choice is basically made for him. No Winterfell; no Stark lord. Time to head north and look for a three-eyed crow.
In the final chapter of the novel, Bran and Rickon Stark part ways. Osha takes Rickon with her, and they head to the East Gate with Shaggydog. Bran, looking out over the remains of Winterfell, thinks to himself that the castle remains, broken but not dead. He imagines that it is the same for him—he, too, is broken but not dead (70.Bran.99).
This is a small, but also really huge, moment for Bran. While he hasn't discovered what his dreams mean or what purpose his life will serve, he has decided to live again, something he's been struggling with for two books. And the quest for Bran to discover the meaning of his life will continue in the north.