1. As an Admin, click on General, scroll down to Classify questions using: and select categories or "tags and categories"
2. Go to RSSFeeds, enable individual feeds for each category.
This displays a list of categories on the main page. If you click on a specific category, you can find the RSS feed for that particular category. What all you have to do is to fetch the RSS feed for this category and show on your website. You can even use headlines animator from feedburner, if you divert the category feed to feedburner.