Visibility means that you make a case private so that agents with a certain role will not be able to view the cases. Sometimes, you don't want everyone at a company to be able to read certain agreements.
You can change the Internal Visibility of cases and templates under the tab Settings.
For templates, you can both set the visibility of the template itself and for cases created from that template.
The default setting is that all templates and all cases are visible for the Agents of the account. By default, Administrators can see other agents' private templates and cases.