From wikipedia's article on API[1]:
> An application programming interface (API) is a connection between computers or between computer programs. It is a type of software interface, offering a service to other pieces of software.[1] A document or standard that describes how to build such a connection or interface is called an API specification. A computer system that meets this standard is said to implement or expose an API. The term API may refer either to the specification or to the implementation.
The server and browser are two different computer programs. The browser understand how to make an API connection to a remote server and then take an HTML response it receives (if it gets one of that media type) and transform it into a display to present to the user, allowing the user to choose actions found in the HTML. It then understands how to take actions by the user and turn those into further API interactions with the remote system or systems.
Because the browser waits for a human to intervene and make choices (sometimes, consider redirects) doesn't make the overall system any less of a distributed one, with pieces of software integrating via APIs following a specific network architecture, namely what Fielding called REST.
Your intuition that this idea doesn't make a lot of sense for a script-client is correct:
https://intercoolerjs.org/2016/05/08/hatoeas-is-for-humans.h...
[1] - https://en.wikipedia.org/wiki/API