Sharepoint - How to get rid of the "Edit links" link in the top navigation (SharePoint 2013)?

The reason there is an Edit Links hyperlink there is because you have the permission to do so, most likely by have the Design permission. If your users do not have that permission level access, they will not see the link.

Also, when creating a subsite, there is an option to display the subsite on the parent's horizontal navigation, you may deselect the option so that your subsite does not appear on the parent site.

If by all means, you must remove that even for people with Design or higher permissions, then you may implement the many suggested CSS. For my implementation, I would use:

<style type="text/css">
    #DeltaTopNavigation > .ms-navedit-editArea
    {
        display: none;
    }
</style>

DeltaTopNavigation is the ID belonging to the div holding all the links of the top horizontal navigation.

ms-navedit-editArea is the class used on a list item for any of the Edit links within a navigation.

The CSS does the following: Look for the element with ID DeltaTopNavigation, then look for the child elements which contains a classname of ms-navedit-editArea, and remove from display.

There might be an easier way, but I was able to hide it by adding the following lines to the masterpage:

<style type="text/css">
#DeltaTopNavigation .ms-core-listMenu-root .ms-listMenu-editLink {display: none;}
</style>

Hope that helps.