The neighbors list is filled with neighbors from the selected spots in the spots list. These are not selected by default and must be selected before any of the search algorithms is executed.
To select all spots in the neighbors list, like depicted above, use the check box on the list header. The square button on top clears all selections (not the list). To clear this list you must clear all the selections in the spots list.
Selected spots from this list are displayed with a blue marker on the map, as depicted below:
The red spot is the one to be moved and the blue ones are the neighbors. Note that in this case there is a clear overlap between spots. In such a case, and if these are somehow related, it might be advisable to select both spots in the spots list.
Reference: What is a "Neighbor"?