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. You can hold down the Shift key at launch to prevent it from re-opening the libraries or opening the startup items.