I think the HTML (or possibly Javascript) example on the Show / hide columns dynamically page is missing the section that shows the
Toggle column: ColumnA - ColumnB - ...
bits. The page source shows this is outside of the "dataTables_wrapper" div so I'm guessing that this done outside of DataTables. I figured this out after pasting the code into my site and being disappointed that the above string didn't automagically appear.
In making sure I didn't miss anything (I had) I followed the Buttons extension for DataTables link above and saw that this should do what I needed, but, it took a google search to get to a demonstration which showed me this is indeed what I wanted. My suggestions would be to have a link to the example in the Column visibility section of the Built-in Buttons page.
And, DataTables is fantastic - thank you!