To subscribe, clients add the URL of the RSS feed file to their RSS reader. If the author of that site updates the contents of that file, when the reader next polls the feed, it will detect the new article and notify the user.
I use Thunderbird as a feed reader. It's more widely known as an email client, but it's actually a really versatile program that supports a pretty wide array of communication protocols.
If you have Thunderbird installed, you should be able to visit my feed, and choose to open the link with Thunderbird.
From then on, Thunderbird should actively check for new posts on some interval.
Generating a feed
Assuming you already have a webserver and some content to share, setting up a feed is fairly easy.
If you're using some platform like Wordpress, your blog probably generated one automatically. If you've written the HTML for your site by hand, or are using a simpler platform that doesn't generate feeds, you just need to create a static file.
A feed consists of some XML, which is very similar to HTML. It will contain some header tags, one or more channels, each of which should contain one or more items. This article explains it in a reasonably simple way.
I use a simple script that I wrote. It converts from a simpler format into a valid RSS Feed.
Each time I update the source file, I run the script again and overwrite my old feed with the updated content.
There are many optional feeds in the RSS specification that you can choose to use, if you want: Many of them are described here along with some of the history of the spec.
Making your feed discoverable
Once you've begun hosting a feed on your site, you'll want to let users know about it. Use a recognizable symbol and link to your feed.
If you want, a site can have multiple feeds for different purposes, just make sure to make it possible to discover them.