This option essentially exposes reports in the report's parent application (not analysis).
The ability to allow "Everyone with this Application" to view reports is based on a set of security role permissions that allow the user to add Global Searches. These permissions are separated by the Application on which the report is dependent (Ticketing for a Ticket report, Projects for a Tasks report, etc.).
If this option is enabled on a report, any user with the matching application (Ticketing for Ticket reports, etc.) will be able to see that report in the matching application, even if they do not have analysis.
I hope this helps!