RSS stands for (depending on who you ask) RDF Site Summary, Rich Site Summary, or my personal favorite, Really Simple Syndication. In a nutshell, it's a way to show you web content that changes regularly (like new blog posts) all in one place, so you don't have to check each individual website yourself.
Here's how it works: every site with one of these buttons has an RSS feed. Using an online tool called either an aggregator, a feed reader, or an RSS reader, you can subscribe to that feed. Then you just need to go to one place to see which of the feeds have been updated. It's sort of like subscribing to a newspaper--a really cool one where you get only the most recent news on the topics that interest you.
RSS readers are an especially useful tool for aspiring writers wanting to keep up with agent and industry blogs. Many are updated daily or several times a week, and it's much easier to take a half-hour to read the new posts all at once through your RSS reader than to check each website separately for new content.
You can even get an RSS reader app on your phone and check your blogs on the go--very handy!
Below are a few of the most popular RSS readers. They each have their plusses and minuses, so I'd suggest checking them out to see which one you like best.
Google Reader (This is the one I use. If you already have a gmail account, there's no need to sign up again, which is nice.)
FeedDemon (According to the site, this is the most popular RSS reader for Windows users.)
Bloglines (I used to use this a few years ago and loved its simple interface. I thought they had shut down, but it looks like someone bought the site and it's back up and running! I might need to check it out again...)
Happy blog reading!