The Mobile Messaging Inbox stores activists' replies to your mobile messages, if they meet certain criteria (explained here).
Go to the 'Start Organizing' menu and select 'Mobile Message Inbox'. Here, you'll see a list of all the activists' conversations, sorted in reverse chronological order, with the newest message at the top. You can page through the conversations, or search conversations by name, email address, or mobile number.
To view a conversation, click from the list on the left side. This will show the activist's reply to your message, as well as some of their core fields and subscription status. You can reply to the message, and view the character count in the lower left. Refresh the page to see any new replies.