The 'Public Messages' stream shows all public messages sent to you by others. Public messages are messages published by fans that are visible for everyone.
Because these messages are public and everyone can see them, it is possible to add a sentiment.
The number next to the 'Public messages' is the total number of posts in the stream.
The meaning 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.