There are several ways to redeem badges. On the one hand, you click on the link here, or you can redeem it in the client. You need a code to get a badge. You can simply copy and paste the code. You already have your badge.
If you still have problems redeeming a Teamspeak Badge, you can have a look at this guide from Teamspeak, where it is explained how to redeem a Teamspeak badge.