In the mailbox My starred you find Tickets that you're assigned to and that you or someone else have starred.

To star mark a Ticket, you click the star so that it goes from "empty" to gold. The agent who's assigned to the Ticket, will then see the Ticket in the mailbox My starred and see the star mark. You can, in other words, star mark a Ticket for someone else and another agent can star mark a Ticket for you - it all depends on who's assigned to the Ticket.

The reason for starring a Ticket, is to mark it as important.