In Carcassonne, what is the point of a Bridge in the expansion set?
Their primary purpose of bridges is to be able to play a tile that you normally could not because of a road. 

You have more options to finish a construction, sabotaging with roads/farms becomes significantly harder.

Second, bridges strongly affect farms. As Nick already pointed out, farms turn out to be massively larger. It's not at all uncommon to have a single, huge farmland that spans the entire map, minus the tiny farms at the edges sealed off by towns obviously. 

On the other hand, invading an already occupied farmland becomes a lot easier than before. 

For example (assuming only base set + first expansion), without bridges, the only way to cross a street in order to invade a farm was with the cloister + single road tile. For that purpose, the Bridges &amp; Castles expansion not only adds bridges, but also several tiles with broken roads to accomplish the same thing.