Shizuku is an open-source project, you can find the source code and more details at https://github.com/RikkaApps/Shizuku.
The birth of Shizuku has two main purposes:
Provide a convenient way to use system APIs.
Convenient for the development of some apps that only requires adb permissions.
As a user, how to use Shizuku?
Rooted devices: Start directly in Shizuku app.
Non-root devices: Follow the instructions in Shizuku app to start the service through adb. Using adb is not difficult, there are many tutorials on the web that can help you learn to use it.
Here's a video showing how to set up Shizuku: