One theory is that the bridge symbolizes hope for a better life. It's the bridge from poor Brooklyn to rich Manhattan. Eddie claims to dream of his beloved niece going to Manhattan and associating with what he considers to be a better class of people.
Another idea is that the bridge symbolizes the narrator, Alfieri. He stands between the Italian and American cultures that clash in the play. Alfieri is a lawyer, right? So, he represents the American justice system. Ahh, but he was also born in Italy, making him equally as representative of Italian ideas of justice. One could say that he's a "bridge" between the two ideologies.
Then of course there's the wildly popular theory that the Brooklyn Bridge is the very same structure referenced by the title of the play. Miller has stated that he wanted audiences to view the play objectively. He wanted people to understand the higher concepts he was trying to get across, rather than just sympathize with his characters. He wanted people to look at the play more objectively, a lot like you would if you were looking down on it from a bridge. For a longer discussion of this, check out "What's Up With the Title?"