One doesn't talk so much about the API of a method, but the API of a class (or a package, or a library). That refers to the methods in a class that are publicly accessible. The javadocs for a class list all the public methods - those make up the API of the class.
Interesting choice of words. It means that the class author defined the API for the class and the author does not want anyone to extend the class and somehow alter the functionality. He used the "final" keyword to enforce his decision that you shouldn't override this method to do something different.
There are some pretty good arguments for making all methods final, though few people actually bother to do it. If you're curious, scroll down to the OO, UML, etc forum and ask about the Liskov Substitution Principle.
A good question is never answered. It is not a bolt to be tightened into place but a seed to be planted and to bear more seed toward the hope of greening the landscape of the idea. John Ciardi