This is fairly simple, and takes less than 5 minutes.
- Download and install Mozilla Firefox (if you aren’t already using it)
- Install the Greasemonkey add-on from here: https://addons.mozilla.org/en-US/firefox/addon/748/
- After restarting Firefox, navigate to this page and install this script: http://userscripts.org/scripts/show/82673
And that’s it! The box should no longer be visible. You may have to refresh your Twitter homepage in order for it to work.
By the way, this will also work in Google Chrome. You do not need Greasemonkey, just click “Install” on the script page and it will be added as a Chrome extension.