Could someone explain how Zotero Integration for LibreOffice works and, in particular, how Firefox participates in this architecture?
Zotero is a solution designed to assist on usage of citations and bibliographic information inside popular word processors. Below I describe the specific use case of using Zotero within OpenOffice/LibreOffice, integrated with Firefox.
Zotero solution consists of three things:
your personal (remote) library, sitting on Zotero servers, which is created when you open an account with Zotero,
your local library, sitting in folders managed by Zotero Extension for Firefox and
a plugin installed in OpenOffice/LibreOffice which has ability to retrieve information maintained by Zotero Extension for Firefox, i.e: has ability to retrieve information from folders maintained by Zotero Extension from Firefox (item 2).
The most common use case is exactly when you retrieve bibliographic information when you are online. Zotero can retrieve bibliographic information from papers (PDF files, in general) which you are viewing inside Firefox. Zotero retrieves information from well known repositories and creates proper bibliographic entries for you, automagically attaching papers you are viewing to bibliographic entries created.
You can also create bibliographic entries by hand, regardless using Firefox online or offline modes.
Regardless you had created bibliographic entries by hand or automagically with assistance of Zotero, you can refer from LibreOffice/OpenOffice without problems, regardless your computer is online or offline.
in order to install Zotero solution, you need to install two extensions, to be more precise.
(a) Zotero Extension
(b) Zotero Integration for OpenOffice/LibreOffice.
Zotero Extension provides you the user interface in Firefox you will be using in order to organize your local library (which is automagically synchronized with your personal (remote) library, sitting on Zotero servers.
Zotero Integration for OpenOffice/LibreOffice allows Firefox to be able to install components in OpenOffice/LibreOffice and installs all necessary packages for you into your running system. You can install manually the Zotero plugin in OpenOffice/LibreOffice and you can also install all required dependencies, if you wish and if you are adventurous, but this Firefox extension automates the entire process, making life much easier.
More information: https://www.zotero.org/support/installation