Need help

Once I have duelled someone on duelling network how do I show it on here.   

It depends if the person said it was for Battle Arena, and from this academy.
In order to get rewarded for dueling, win a duel/duels for Battle Arena.
http://www.duelacademy.net/f21-battle-arena

Then have the person who losses confirm. Simply by posting something that verifies that you won.