What does DELIMITER // do in a Trigger?

It changes the statement delimiter from ; to //. This is so you can write ; in your trigger definition without the MySQL client misinterpreting that as meaning you're done with it.

Note that when changing back, it's DELIMITER ;, not DELIMITER; as I've seen people try to do.

In SQL you close each statement with a delimiter, which is by default a semicolon (;). In a trigger you need to write multiple statements, each ending in a semicolon. To tell MySQL that those semicolons are not the end of your trigger statement, you temporarily change the delimiter from ; to //, so MySQL will know that the trigger statement only ends when it econunters a //.

Add an example:
When working with mysql shell command, we used ; delimiter to close each statement. However, in case, we want to build the store procedures and triggers, we need to add the semicolon ; in these statements also.

delimiter //
create trigger log_students after insert on students
for each row
  begin
    insert into log_students(change_by, change_at) values(USER(), NOW());
  end//
delimiter ;

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.