Click on the little box to the right of the message so that it's checked. You can check all of them by checking the one at the top. Then click on the proceed to message button. It should come up with a little pop up that gives you the option to save messages to a txt file or some other formats. Download the message(s) and save them in your documents or wherever you'd like to keep them.
The inbox is pretty small, so I have to keep mine cleaned out as well.
I hope that made sense.