In order to use your badge containing your username, rank and points on the Forums signature, you'll first need to get your badge icon.
In order to do this, visit your Dashboard, then click on your profile picture on the top right and click on the Visit My Profile link.
While on your profile page, click on the Get Badge button below your Rank Progress meter. This will initialise the popup menu where you can select the code for the static badge. The Forums don't currently support <script> tags so you will need to update it manually with a new static link every time you reach a new rank.
After you copy the <img> string, visit the forums at this link.
On the top right-hand side, you will need to click the Settings cog, then Preferences. On the right, navigate to Signature Settings. In the text-box on this page you can paste your static <img> link to your badge together with any other links you'd like people to visit, such as your Twitter, MySpace, etc.
You can now test out the signature functionality by posting something on the forums! The signature will appear right underneath your post.