In the Elements 13 Editor, what is the File (menu)> Export cmd used for?

The greyed out options in Elements editor File menu cover functions not offered by the software itself, but as you suspect rightly, by additional add-ons and scripts.

I don't know when the 'export' function can be used and for which scripts. I do know that the 'automation tool' option is only used by many external tools running scripts. Otherwise, it's greyed out.

I had a need to export some of the layers into another file so that I could edit them, such as cropping and transposing, without affecting the other layers.

That's something you can do manually in Elements, but you are expecting Elements to offer all the tools of Photoshop. One voluntary limitation in the features is anything relating with batch editing and customization.

For ex-Photoshop users downgrading to Elements, I recommend using external tools like Elements+ (my choice) or Elements XXL which restore hundreds of features of Photoshop. For instance there are scripts to save layers as individual files. That will get you started rapidly. Otherwise, other Elements users may suggest one of the many ways to achieve the same goal... that may be slower or less obvious than with the myriad of options of Photoshop.