The advice isn't suggesting that the app should be stateless - it's suggesting that the resources in the app should be stateless. That is, a page called "www.mysite.com/resources/123" will always represent the same resource, regardless of which user is accessing it or whether they're logged in or not.
(The fact that you might deny a non-logged-in user access is a separate issue - the point is that the Uri itself doesn't rely on user-specific data to work.)
For example, the kind of sites that break this rule are those where you navigate to a product page, email the Uri to your friend, and on clicking it they see a message along the lines of "I'm sorry, your session has expired" or "This product does not exist" or similar. The reason this happens is because the Uri includes something specific to the user's session on the site, and if a different user tries to use the link (or the same user at a later time), it's no longer valid.
So, you will always still need some form of state for your application, but where that state is implemented is the important factor.
Hope that helps shed a little light!
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