Why is the Links Manager visible?

If you install WordPress using the GoDaddy cPanel then GoDaddy will automatically install a number of Must Use Plugins in your installation. These plugins are invisible and you can’t disable them by normal means. One of these plugins is enabling the Link Manager.

Aside: If you haven’t worked on your site yet I recommend installing WordPress manually. The plugins GoDaddy adds will add unnecessary features and sometimes conflict with other plugins.