TL;DR: To store files from App Engine to Cloud Storage, please use the Cloud Storage client libraries available in different languages. Each one will have its own open source repo and documentation. Those libraries can connect your app to Cloud Storage, usable outside of App Engine and any other Google Cloud compute platform.
The libraries in this repository replaced the original App Engine Files API deprecated in 2013 and represented the original libraries for Cloud Storage created just for App Engine users. Since then, the official Cloud Storage client libraries launched in 2016 (linked above). As a result, work ceased in 2017 on these libraries, which were deprecated in 2019, and are no longer recommended. For archival purposes, here is a link to the documentation for the Python version of this library. It will be removed at some point in the near future. (The Java documentation was already removed in 2021.)