On the "Organization Settings > Member Privledges" Page, I see this:
I want to select the second option, thus allowing members of my Organization to make only Private Repos (not public ones), but that option is greyed out. I'm unable to find any description of why I can't select that option? Can anybody explain why that option cannot be selected?
You need to have a paid plan on your organisation in order to be able to create private repos in them. I suspect the option will be available when you're on a paid plan.
This sounds like a billing issue. Get in touch with private support at https://github.com/contact . Due to privacy reasons, no staff members here will discuss billing details.