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How does it work? I am on OS X 10.8 Mountain Lion.

There is one big caveat: the developer must have made the last compatible version available.
Then: http://support.apple.com/kb/ht5919

If I go to the purchased tab in the App Store, and click on download for a given application, I get a popup which states "Download an older version of this app? The current version requires OS X 10.9 or later, but you can download the last compatible version."
I don't know if it works for apps you have not yet purchased.
 
I agree with you on principle. iOS 7 is a significant change for most people, and Im sure many people have upgraded without actually realising how significant update it was. Apple really should allow a downgrade, however thats not the way apple does things. On top, well have to live with iOS 7 now, its not going anywhere, and everyone will have to adapt it at some point.
I don't have to live with iOS 7, I'm enjoying Android.

iOS 7 is not a "significant" update, it is a very bad reskin.
 
I tested it with iLife and iWork in 10.8.5 yesterday and it works there. It does not seem to support 10.6.8. If it's not working for you with 3rd party apps, it may be because the developer has not opted to participate in this program.
EDIT: I just tested with 10.7.5 and it does not work with that, so it appears that this only works in 10.8.+

Does this not work anymore?

I’m on High Sierra using Pages 7.1. I think they highest version of Pages available for High Sierra is 8.1. But nothing happens when I press update except a little spinning circle.
 
Does this not work anymore?

I’m on High Sierra using Pages 7.1. I think they highest version of Pages available for High Sierra is 8.1. But nothing happens when I press update except a little spinning circle.

My experience is that it doesn't work anymore. I tried to reinstall Pages/Numbers/Keynote on a 2009 iMac that I wiped and found out that even though these programs appear on my purchased list, they would not download and I was not offered to download an older version.
 
My experience is that it doesn't work anymore. I tried to reinstall Pages/Numbers/Keynote on a 2009 iMac that I wiped and found out that even though these programs appear on my purchased list, they would not download and I was not offered to download an older version.

I got it to work. I had to trash Pages 7.1 AND Pages ‘09 and then from the purchased list I clicked on “Install” for Pages. I got the message “the current version requires OS X 10.14 or later, but you can download the last compatible version”. I got Pages 8.1.
 
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