I understand how to use the Bridges (sort of), but I don't understand the strategy/purpose. In almost every expansion set the special rules or powers of the pieces makes sense to me but why would I play a bridge piece? If you are playing with the King set I understand there is a bonus for the longest road but other than that, what is the point of a bridge?
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 & Castles expansion not only adds bridges, but also several tiles with broken roads to accomplish the same thing.