When you go to your FCOM/QRH and find your Landing Distance Without Auto-Brake, the numbers you get are for (Actual Landing Distance) a 50ft over the threshold, No Flare and Maximum Brake. (I believe this are minimum demonstrated figures).
The 1.60 rule, allows for smoother landings, flare included, low on brakes, and longer touchdowns.
If you compare those Landing Distance Without Auto-Brake tables with the Auto-Landing Distance ones (with auto-brakes medium), you will find that the later ones are roughly the first ones (Actual Landing Distance) multiplied by 1,60.