Im new here, just became member, what is the activity box for?

It's the forum's version of Facebook statuses, or posting a new Tweet.

It's your status wall posting box and below that your status wall. Statuses also appear in a public feed.

If you click the Activity link at the top of the forum ^



It takes you to the (public) feed of peoples' statuses. If you go to your Activity page (mentioned in your post) it shows all your own statuses and replies. If you go to another person's Activity wall, it shows their statuses and replies.