A private Category allows you to create Products and "hide" them from the Public. The Products would only then be visible to staff or to customers that you provide a direct link to.
This is useful if you'd like to create a private Product to be booked by one customer, i.e. you could create the Product and Session then provide the customer with the link to this Product to book in.
Another reason you might use private Categories is if you wanted to integrate a single Category on your website without displaying other Category tabs.