To build an Android app using an API, developers can start by signing up for a RapidAPI account and accessing the Aztro API. They then need to subscribe to the API and test its endpoint, which expects a few parameters such as the sun sign name and day. The app's main activity is where the API call is triggered by a button click, using Kotlin coroutines to handle the asynchronous execution. The API response is then converted to a JSON object and used to display the horoscope prediction on the app screen. The app also includes a dropdown spinner for selecting the sun sign and a result view for displaying the prediction. After building and testing the app, it can be deployed to the Google Play Store.