RSS stands for Really Simple Syndication. It's an XML-based format used to share website content that can be read using software called RSS readers or aggregators. An RSS "feed" is an XML (XML is a programming markup language that your computer browser reads) document that includes specific content. RSS readers check the content in a feed at selected time intervals and display any new content in the reader. Some readers will notify you when an update is available and others require you to check the reader to see the updates.
A benefit to using an RSS reader is that you can receive updates from websites without having to visit each site. In the case of the Legislature's website, you can receive updates on a number of bills without having to visit the page for each bill separately.
Many RSS readers are available, some of which are free. There are both web-based readers that allow you to view feeds online and those that are desktop software that you download and install on your computer. Many newer browsers are also RSS-capable and they will display the feed content directly in the browser.
For more information, see About Washington State Legislature RSS Feeds.