This will be brief.
Inbox zero is a valuable thing to maintain. Roughly promoted around the web as having an empty inbox.
An email inbox collects a few things:
- junk
- automatic mail sent to you
- personal mail sent to you
- work sent to you
- (maybe - work you send to yourself because that's the best way to store information for now)
- Old as all hell (or other friendly name)
- 2014
- 2015
- 2016
I read the advice as:
If you still have unresolved emails from 2015 in your inbox then keeping emails in your inbox isn't causing them to get resolved. Accept that, get a clean slate, and move on.
Make a folder called "old inbox" and put all your old emails there. Now you have an empty inbox! The costs of putting your old emails out of sight are less than the benefits of keeping an empty inbox going forward.
I do not believe the intention of the advice given is that emails in your inbox that you feel require some reponse, but that you don't see how to deal with completely in a few minutes, should be archived and forgotten. (Perhaps I misunderstood?)