To understand the use of "SET SERVEROUTPUT ON" I will take an example

DECLARE
a number(10)  :=10;
BEGIN
dbms_output.put_line(a) ;
dbms_output.put_line('Hello World ! ')  ;
END ;

With an output : PL/SQl procedure successfully completed i.e without the expected output
And the main reason behind is that ,whatever we pass inside dbms_output.put_line(' ARGUMENT '/VALUES) i.e. ARGUMENT/VALUES , is internally stored inside a buffer in SGA(Shared Global Area ) memory area upto 2000 bytes .
*NOTE :*However one should note that this buffer is only created when we use dbms_output package. And we need to set the environment variable only once for a session !!
And in order to fetch it from that buffer we need to set the environment variable for the session . It makes a lot of confusion to the beginners that we are setting the server output on ( because of its nomenclature ) , but unfortunately its nothing like that . Using SET SERVER OUTPUT ON are just telling the PL/SQL engine that
*Hey please print the ARGUMENT/VALUES that I will be passing inside dbms_output.put_line
and in turn PL/SQl run time engine prints the argument on the main console .
I think I am clear to you all . Wish you all the best .