How are permissions assigned for approval?

Approvals are handled by regular SharePoint permissions and settings. In order for a manager to approve a request that user need to have that permission level set (Approver in this case). This can either be set on the parent (host) web or on calendar list itself. The former is recommended as there is no easy built-in way to handle permissions on individual lists on app-webs unfortunately.

In order to set permissions on indivdual lists, you need to click on List Permission on the list settings page. The easiest way to get to the list settings page is by going to a list view of the list, clicking Modify View then clicking on Settings in the breadcrumb title link. This process would ideally have to be repeated for the calendar list (LeaveCalendar/Lists/Events), the WorkflowHistoryList (LeaveCalendar/Lists/WorkflowHistoryList) and the WorkflowTaskList (LeaveCalendar/Lists/WorkflowTaskList).

In addition, please note some of this information and then some you can find under the Support page in the add-in.