Find code samples that use that API, then study them.
Unlike unit tests or reading the API inline documentation, code samples showcase an entire use-case for that API, start to finish. You can see precisely what the inputs and outputs are, and you can usually understand from the context what it is supposed to do. Even more complicated APIs can be deciphered this way, since you see the entire picture. It also makes sense - you are not concerned with internal library implementation details or how to test it, you are concerned with how to use the API, so you look at other programs that do that.
Where to find code samples featuring usage of the API?
Naturally the only code you will be able to find is open-source code. A good way to find relevant samples is to use a search engine for open-source code, such as Github Search or Koders. Another way to try and identify other project from the library's author, which are likely to use it.
Finally, you could also use sites such as Stackoverflow to ask for a code sample for a specific use-case, e.g. "how do I transmogrify zebras from low-quality to high-quality with the ZebraTransmogrifierHD library?"... or you could find existing questions there that ask for how to accomplish things.
What are the disadvantages?
The problem with looking at a code sample, as opposed to finding a tutorial, is that some lines might not be immediately understood by you, and yet you will be eager to try the API so you will reproduce the entire sample in your own code, including these lines. Having lines in your own code that you do not understand is a very bad idea!
What I do in those cases is to tinker with those mystery lines - what happens if I move them around? Remove them altogether? Change the parameters? etc.. This usually helps.