Any chance this is on a Mac system with Snow Leopard (OS 10.6) installed on it?
In the screen shot you are showing is from Table View. In order to have the "handlebars" around an object or field you need to be in Form View. In table view all you really can do is select "Modify" and add fields to your table or modify the contents of the field. Keep in mind that even though you have inserted a field to the layout, you have to go back and add the same field to your table. I'm not sure why your fields disappear, you are working with a excercise DB. In my experience that excersise tables and DB work a little funny. (or several times over the course of the filemaker training series)
Keep in mind that even though you have inserted a field to the layout, you have to go back and add the same field to your table.
You cannot add a field to a layout without it first being defined in one FileMaker table or another.