The browser had a built-in RSS button that would display in the browser location bar when any website you're on had an RSS feed available. Clicking the button would then take you to the RSS feed for that web page
How would this work? Do websites with rss feeds normally publish the url to that feed in some standard place? Are there any third party extensions that do it?
Most readers will also do this auto-discovery for you. So typically you can just paste the page or article URL and it will find the feed.
Of course the extension is nicer because you don't need to guess and check, you get a quick indicator if there is a feed or not.
Personally I use Want My RSS because I like the preview which then lets me know if it is a full-text feed or just summaries. This is also Firefox only. But extensions for other browsers are available.
Do a View Source on the site's frontpage. You might see some HTML for "application/atom+xml" or "application/rss+xml". The URLs associated with those hrefs will be for the ATOM and RSS feeds.
If you search for one of the following in the HTML source you'll probably run into the feeds:
rss
atom
feed
json
Look for a syndication page on the site. It should have links to the feeds.