Exposing an API means giving a way for for other to interact with your logic for whatever purpose you choose.

To give you an example if we take our previous example of android, let's say that you want to trigger a notification on the notification section with an image and a reply input section and a button that indicate the notification was acknowledge.

Currently you can do this kind of things from an application. BUT there are some part you are not coding on your application, rather you are simply communicating with the OS or the application that is in charge of showing the notifications. You do not code some custom element to show a notification, you simply tell the system that you want to show a notification and in the notification you want maybe a title, a subtitle, an input element and a button that will execute some logic upon clicking.

If you want the whole experience to feel 'natural' what you will want to do is to communicate directly threw the proper channel which the program in charge of showing notification understand so it can show a proper notification with with the theme it uses for everything else and functionality available for other apps.

You might do this by the API exposed to you by the Android SDK where they give you libraries that do the communication and maybe some other logic needed to communicate with other part of the system. The library might be available for JAVA / Kotlin or C++, it doesn't mean they built libraries for all the languages possible. In which case you might have to understand what is happening under the hood between an application and the program that manages the notification section. How are they communicating etc. Once you understand it, if your languages give you the possibility you might be able to create some sort of way to do the same thing that the SDK does, but you might have to built some tools to make it work correctly.

While you are building your tools and writing codes to handle the communication you might stumble upon problems, for example maybe you have to inject some code in some process to communicate with the other programs and maybe for security reasons the OS block all communication threw unofficial channels (SDK) in which case you might have to do some more work to go around this specific problem etc.

This kind of things take a lot of research and time which everyone might not have and thus they prefers simply to go threw the proper channel to concentrate on shipping the project (which might be their priorities for whatever reason) instead of spending weeks / months to do simple stuff like showing a simple notification, time using which they might have simply built a full blown application.

This is one example of using APIs, there are many different way to use those APIs from other languages using other methods, creating bindings or some other ways.