What does DELIMITER // do in a Trigger?

Simple sets the end of statement delimiter (; semi-colon in standard, default SQL). Changing the character could be useful if you want to use ; in your SQL, or you are using embedded SQL (where it might lead to confusion). similarly the // in your example could lead to confusion in embedded SQL, or you might want to use it in your SQL. So imply use DELIMITER to set the delimiter that is appropriate for your application and needs.