Crane fits in with a whole category of people who are described as "spiritual but not religious." His spirituality seems roughly in line with the twin 19th century currents of British Romanticism and American Transcendentalism. He thinks the old religions have passed their prime; their myths have become faded and ill-equipped to deal with modern life. In "To Brooklyn Bridge" (and the larger poem containing it, The Bridge) he tries to build up a new mythology that's largely centered around urban life and the American city. Think of Simon and Garfunkel's classic lyric: "The words of the prophets are written on the subway walls." That's an idea similar to what Crane is proposing here. Not surprisingly, The Bridge was an inspiration to the Beat writers, who shared a sense of deep spirituality and a distrust of traditional institutions like churches and synagogues. You might compare "To Brooklyn Bridge" with Allen Ginsburg's famous poem Howl, also about life in New York City.
Crane's choice of the bridge as a quasi-religious symbol is arbitrary and really reflects his casting about for something in the world to replace the void left by religion.
"To Brooklyn Bridge" expresses the modernist idea that art and religion are one and the same.