To understand his dreams more
Bran wants to go north of the wall because he wants to get more understanding of his visions.
To find the three eyed Raven
Bran wants to find the three eyed Raven he sees in his dreams.
Jojen convinced him
Jojen is the one who convinced Bran that everything happened to him for a reason and that he should go north of the wall.
Everything Jojen said was true
During a debate about whether to go north or not Bran told Osha that everything Jojen said was true and that he believes he has to follow him. (See Why did Robb Stark die in Season 3?)
Bran thinks Castle black is no longer safe
Bran asked Osha to take Rickon to Last Hearth instead of Castle black because he believed it might be no longer safe after he saw the Wildlings coming close.
Bran believed it’s all planned
Jojen convinced Bran that everything that happened to him was planned and that a part of the plan is to go north. (See Game of Thrones Season 3 Summary)
Bran knew Jon isn’t there
While Bran was using his Warg powers he realized that Jon Snow isn’t in castle black as he saw him running from the wildlings.
Bran Knew he is special
Bran realized that his dreams are visions and that he has special powers. This made him believe that the rest of his visions will lead to something worth finding.
Why is Bran Stark going north of the wall?
In Season 3 of Game of Thrones, why is Bran going North of the wall?