Short answer:
NO need to install MS Office on the server,
"NO" need to install MS office on clients
Let me try to explain this briefly.
OpenXml
is a file format: compressed package containing xml files.
OpenXmlSDK
is a wrapper for this file format: class that allow you to create package, or modify it's xml content, it doesn't require MS office, and it's the real advantage in server side application.
So no need to install MS Office on the server.
Then you could imagine that clients are able to create OpenXml files on the fly and download them on their computers. If they have not MS Office installed, they won't be able to open the created files, but the files are still valid.
So "technicaly", clients do not need to have MS office installed to interact with your server application.