On archived messages screen you can see your chat history with those advisors who are either inactive or Away. You can also delete messages from here to get back credit if you have unanswered messages. Click on trash can icon next to unanswered message to get back credit.
On Live Psychic Chat iPhone App you can access this screen from More -> Archived Messages.
On Live Psychic Chat Android App it can be accessed from More -> Archived Messages.
On Live Psychic Chat Web App you can access it from My Profile -> Archived Messages