For instance, you can use Twitter via their website OR you can use a third party application that asks twitter for the data it publishes, then the application uses that data to show you tweets, users, etc. It can also, with your and Twitter's permission, send tweets, update your profile, etc.
So BC could publish these APIs and developers could build apps that might logically look like this:
App: Using this BCID, tell BookCrossing that I (a user) found this book.
BC: I have recorded that you registered your capture.
App: This user is releasing this book.
BC: I have recorded that user has released this book.
And so on...
APIs mean that organizations who want to build a great website don't also have to build a great app. They just build a service and let others, who know a lot about apps, build the apps. The original website doesn't have to change at all. The services (APIs) are built separately.
Websites can also charge for the data, so that they can afford to supply the data and provide the services that share and update it. That's what Twitter does. Some comes from free, but other data comes at a cost. It's a win-win.