A "Page", in Google+ parlance, is a sort of virtual account that is managed by one or more real user accounts. Unlike the similar feature on Facebook, pages have no separate log-in; you log in with your user account, navigate to the Page you want to manage, and then click the "manage this page" button which shows up if you have permission to manage it.
When you are "managing" a Page, you see everything on Google+ as if you were a different user. Pages have their own circles, streams, posts, comments, profile, and so on.
The way Google+ keeps track of how you are viewing content (i.e. whether you are using G+ as yourself or as a Page) is to insert the Page's user ID into the URL. That is, G+ does not appear to keep this information anywhere else, such as in a cookie or in the data it maintains about your session.
The last two URLs will not work unless you have permission to manage the "Pip" Page, but you can substitute the ID of any Page for which you do have management permission by replacing "108982124935947195046" with that Page's ID if you want to see what you get:
|the public profile of a Page ("Pip"):||https://plus.google.com/u/0/108982124935947195046/posts|
|managing the "Pip" Page:||https://plus.google.com/u/0/b/108982124935947195046/dashboard/overview|
|the public profile of a real user (Woozle):||https://plus.google.com/u/0/102282887764745350285|
|a post by Woozle:||https://plus.google.com/u/0/102282887764745350285/posts/J7iqF3Mpfrs|
|viewing Woozle's public profile as "Pip":||https://plus.google.com/u/0/b/108982124935947195046/102282887764745350285|
|viewing Woozle's post as "Pip":||https://plus.google.com/u/0/b/108982124935947195046/102282887764745350285/posts/J7iqF3Mpfrs|
Note the insertion of "/b/108982124935947195046" immediately after "/u/0" in order to switch from viewing an item as yourself and viewing it as "Pip".
For some reason, G+ has made it very difficult to switch, via their user interface, from user-view to Page-view while looking at the same thing -- even though this is a fairly trivial operation if you know how to do the URL modification. Such switching is necessary when, for example, one comes across a post where a comment from a particular Page would be more appropriate or meaningful than a comment from one's main identity.
Because your identity (i.e. whether you're viewing G+ as a Page (and which one) or as your main account) is embedded solely in the URL, you can't just switch identities and then paste the URL of the item you're trying to view because that switches you back to whatever identity you were using when you copied the URL.