The Brooklyn Bridge, as has been discussed by people far more scholarly than me and by people with teredo issues has been a cultural icon since it was built. On a personal level I can say that I would probably have drifted into structural engineering even if I had never seen it, but having read about it far more than I should have when I was 16, it was probably inevitable that I would attend Washington Roebling's alma mater.
The question comes up: why is it famous now? It was not the first record-setting long-span suspension bridge (although it did break the existing record by 50%, which is an achievement in any era), it hasn't had the longest span since 1903, and the hypertrophied NYC self-promotion machine has not made the newer and far bigger Verazanno Narrow Bridge famous as anything other than a pain in the ass for people in Staten Island and Brooklyn.
The answer, in my opinion, is a combination of two factors. First, the bridge is beautiful in its structure. There is a small amount of applied ornament - most obviously, the cornices and gothic arches of the towers - but most people who see it in person fixate on the appearance of the cables. Structures that are beautiful in themselves are rare, and this one is in a location that makes it incredibly prominent.
The standard view from the side used for bridges doesn't usually show the Brooklyn Bridge very well as it emphasizes the heavy masonry towers (click on the picture twice for full-size):
A modern high-res photo gives a better idea of the visual effect of the vertical suspender cables and the diagonal brace cables, even with the Manhattan Bridge standing in the background making rabbit ears over the Brooklyn's head (click on the picture twice for full-size):
It's pictures from the walkway that really give the full effect:
The second reason is less visceral but more obscure. The Brooklyn Bridge was completed in 1883 at a critical moment in engineering history. The State of Liberty - the first tall steel frame in the United States - wouldn't be completed until 1886. The first steel-skeleton skyscrapers would be completed in New York and Chicago in 1890. The Brooklyn Bridge was the last large suspension bridge with masonry towers that visually (and in structural analysis) provide a rigid anchor for the flexible cables and deck. We know now, looking back, that this was the both the last of the old-style bridges, with six years devoted to masonry work before the steel superstructure began, and the one of the first demonstrations of the possibilities of steel construction.
But, hey, I just like it.