Atom and RSS are two similar methods of web content syndication. The idea of feeds is not to check all web sites of interest one by one, but automatically collect updates from them with one tool called "feed reader" or "aggregator", so feeds can be considered as an advanced alternative to bookmarks. Subscribing to feeds really helps to save time and stay informed. Blogger supports both Atom and RSS feeds.
The truth is that nowadays web browsers can work directly with feeds, so a special program is not really needed. For example, Firefox shows available feeds as an icon in the end of the address field:and Internet Explorer - in the Command Bar: Click the icon to open available feeds from the web page, choose the one you prefer and that will take you to a subscription page. Firefox can save a feed as a "live" bookmark which references not just a website at whole but each article individually from it. IE puts it under tab "Feeds" in the "Favorites" window.
While it is possible to subscribe to the feeds directly from a web browser it is still can be good for the sake of convenience to add them to a web page. Blogger has some variants of the feeds - Blogger Help describes them in the "Blogger Feed URLs" article.
For more information a great description of Atom and RSS can be found in Wikipedia.