To assign a ticket manually to a particular staff member, open the ticket in edit mode. Out of the box, the Assigned Staff column is on the "Staff" tab. This is one of the many things you can modify, so the Assigned Staff tab may exist elsewhere in your implementation of the program. In either case, open the ticket for editing prior to manually assigning the ticket.
Once opened, navigate to where the Assigned Staff column appears. You can fill in one or more users for the Assigned Staff column using the standard SharePoint "people-picker" functionality.
You can also assign tickets using the function. See that section for more details.