If this is checked, EagleFiler will remember which libraries were open when you quit it, and it will re-open them the next time EagleFiler is launched.
In addition, you can put aliases to your favorite libraries in the folder:
/Users/<username>/Library/Application Support/EagleFiler/Startup Items
so that EagleFiler will always open them at launch. (See How can I open the Library folder?.) You can hold down the Shift key at launch to prevent it from re-opening the libraries or opening the startup items.
If EagleFiler is closed, and you don’t want it to re-open a certain library at the next launch, you can remove that library’s alias from the folder:
/Users/<username>/Library/Application Support/EagleFiler/Open Libraries