Why is Office required for multiple file uploading?
I don't have Office, is there something I can install that will enable this feature?
There is an ActiveX control that is used to support multiple file uploading - that ActiveX control is installed with office.
@Kirk: Depending on your SharePoint version it should be sufficient to either install SharePoint Designer 2007 or 2010.
Edit: Forgot to mention that SharePoint Designer is a free download from the Microsoft Website.
Have a look at this recent question 'What are the client side requirements to enable the “upload multiple files” link?'
You can also upload multiple files using WebDav (Explorer View). Office is not required for that. See this question for requirements.