Maybe the name gives it away, but the 'All messages' stream shows all messages in the selected brand. This includes your own messages and replies. You can set this stream in two ways: newest messages first and oldest messages first. The left screenshot below shows the stream when it's set on newest first, the right one when it's set on oldest first.
The version with the oldest messages first shown below is a special edition. The timer that is shown is a customer feature that admins can request. For example, if you have a goal to answer messages within an hour, you can let us set the timer to an hour. If the oldest message in the stream is older than an hour the red bar will appear. If not, the red bar will be green saying you're safe. The text in both bars is fully customizable.
Do you want the timer in your brands as well? Let us know via email@example.com.
The meaning and use of the three icons in the upper right corner of the stream window is explained here. They allow you to edit the stream and select one or several filters.
The number next to the All messages is the total number of posts in the stream. The more filters you have activated, the lower this number will be.