There's no way to get at this activity other than through a settings shortcut. The code for it can be found here on Github, and if you search the repository for the
NotificationStation class you'll find all the entry points for it: in this case, only the shortcut intent filter in the manifest. If you examine the code you'll also see there's no way to remove old notifications from the list: as you can see in the screenshot, the greyed-out notifications are those you've dismissed from the 'normal' notification tray.
However, this class is mainly intended as a demo. Android 4.3 adds support for any application (with the appropriate permission) to access and display the list of notifications. For example, a third-party launcher might want to show them, an app might want to show all notifications in a pop-up window iOS-style, a live wallpaper or daydream could respond to them graphically, or a utility might analyse your phone use from the notifications (something like Tasker could trigger actions based on notifications appearing or being dismissed).
Because this is a new feature, and 4.3 isn't on many devices yet, it'll be a while before third-party developers invest the time needed to offer these features, but you won't have to wait long.