The gist:
RSS is most commonly used to keep users of news websites, blogs, vlogs or podcasts up to date. If you see this sign 
you click on it to subscribe to the feed on the website. Updates will be delivered to a feed reader of your choice. You can the see if there is new information on the sites you frequently visit without having to look at each one of them separately.
Why users want to use this:
Because it saves them time and effort. Once they have made the choice to get a specific content delivered to them, they will receive all the updates that are happening without having to check the site all the time.
Why publishers / organizations want to use this:
Because it keeps them in touch with their users. More and more people want to subscribe to updates without the danger of receiving spam in their email. RSS makes that possible. It is also a simple and cheap way to get updates to users.
Tech specs:
More information on RSS can be found on these sites: