Your inbox in the app has two main features. First, you can view your text messages in a conversation format, (as if you were using the text app in your phone) and all of your recent activity such as texts, listing views, and new leads. You can filter out just your inbound messages on the 'Inbox' tab at the top, and view all activity in order with the 'Activity' tab.
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If you tap on a text message (denoted by the red & white chat bubble icon) you will see all of your text history with your contact in one place. Inbound messages are anchored on the left, and your messages are on the right.
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On the activity tab, you can tap on any activity that is not a text message, and it will take you to their contact details/timeline.