In all honesty I can't say I know a whole lot about what those specific sites are doing and how their system is setup and designed to work.
Obviously you can leave the trailing slash in a URL of a folder off and it will still load, but as a redirect (which is slower) and typically you'll see the slash added once the request has succeeded. But other than that, a URL with no trailing slash typically means it'd be pointing to a file rather than a folder. But that doesn't mean you can just go take extensions off your pages because they'll load as plain text in a browser rather than a webpage.
I've heard of .htaccess 'tricks' that can redirect to index pages without adding a trailing slash but I don't really know much about that. Other than those few things I can't really tell you much. Obviously avoiding the use/need of sub-directories is a better way to go and so perhaps there is some 'trick' to getting a file (lacking an extension) to somehow redirect or load content which would give the desired result.