RSS feeds let you see the latest updates in a project from a feed reader. Each project has its own authenticated RSS feed. You can find it at the bottom of each project page.

You'll need a program that can read authenticated RSS feeds to read your Basecamp updates. On a PC you can use Feed Demon. If you're using a Mac, you can use Net News Wire. If you want to get upates on your iPad, you can use Feed Hopper.