The Forms-folder, as the name and the content suggest, contains all the
- Forms (for displaying [DispForm.aspx], editing [EditForm.aspx], uploading [Upload.aspx], etc.)
- Document Template (by default an empty template.dotx)
- Views (the All Items View [AllItems.aspx], Thumbnails View [Thumbnails.aspx] and some helper Views [repair.aspx, Combine.aspx]).
Let's say you create a new View for your library and you call that view MyCustomView, then you will find another file in the Forms folder called MyCustomView.aspx containing the definition of your View
The Forms-folder is usually hidden from both, viewing it on the web and in the file explorer and therefore won't be copied when you copy all documents to a new location. I guess you have the Show Hidden Items option activated in your file explorer and thus, you were able to copy the folder.
It is safe to delete the Forms-folder from the subfolders at the NEW location. The Document Library where the files reside now, will use the its own Forms-folder located at the root of it.