The gist is that you start by writing example client code demonstrating small but useful interactions with the API, including explanatory comments. They won't work yet, because the API hasn't been implemented. But you're already in a position to talk them over with your team and start thinking about user experience issues. And they give you a nucleus around which you can start building high-quality user documentation.
And they're your automated end-to-end acceptance tests.
Once things look good (not perfect), then you can start implementing the API. For this portion I don't necessarily do TDD, because I remain unconvinced that writing unit tests first is inherently better than writing unit tests afterward. But I might. Depends on my mood. But I do keep running the walking skeleton tests, which serve as my primary feedback mechanism for evaluating implementation progress and success.
The more important thing is that I can start stepping through those walking scenario tests and seeing how well the interaction actually works in practice. And I can use those observations to help inform an ongoing conversation about the API's design and functionality. I can also get quick feedback on whether some aspect of the high-level design isn't going to work as well as we had hoped. Ongoing updates to the design are memorialized and communicated by updating the walking skeleton tests.