After applying the AJAX Edit-in-Place server behavior according to the instructions described here, the edit-in-place widget in browser should look as it follows:
When mouse-hovered, a pale yellow background shows, as well as a tooltip that notifies you about the edit-in-place feature:
After clicking it, a textarea it displayed where you can type the new value:
Use the Save/Cancel links to save the new value in the database or to abort the change.