In general, the first thing you need is something called a news reader. This is a piece of software that checks the feeds and lets you read any new articles that have been added. Many aggregators are separate, "stand-alone" programs; other services will let you add RSS feeds to a Web page.
Browser-based news readers let you catch up with your RSS feed subscriptions from any computer, whereas downloadable applications let you store them on your main computer, in the same way that you either download your e-mail using Outlook, or keep it on a web-based service.
If you click on the RSS button you can subscribe to the feed in various ways, including by dragging the URL of the feed into your news reader or by cutting and pasting the same URL into a new feed in your news reader. Most sites that offer feeds use a similar orange button, but some may just have a normal web link.