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.
The bridges help to significantly expand farmland, as farms continue underneath the bridges. Using them, you can make really quite massive farms, which means a truck load of points at game end.