How do you start your game with spring framework?
If you want to understand how LibGDX is structured it's probably best you create a project with the libgdx setup and try to fiddle around with it a bit.
When you generate your libgdx project, you have one main project containing your game code and you have separate projects for each target environment (desktop, android, web etc.). You can find *Launcher.java files in the target projects.
This is the startup code from my DesktopLauncher.java
LwjglApplicationConfiguration config = new LwjglApplicationConfiguration();
config.width = 800;
config.height = 600;
new LwjglApplication(new Matcher(), config);
In this case Matcher is my main class extending ApplicationAdapter. I think this should be what you need to start your LibGDX application.
Just one thing to consider if you want to use spring. I am not 100% sure but I could imagine that it won't work (or will be a pain to port) with android and web projects. If you don't care about these platforms then you should be fine, otherwise I would investigate if you can even use it and if yes how hard it will be. Could save you some time in the long run.