Making a drastic simplification: let's say I used that axe ten years ago to butcher an animal and someone found the axe today, along with the carcass. Date-by-association would make the axe ten years old, more or less.
Not trying to be argumentive here -- just wondering how they can be so sure they aren't dating the actual rock material itself, rather than the axe-creation date?
So we have an axe and a carcass buried together, and we have a second axe (call it Axe #2).
The dating mechanism in this case only works on organics, so they are dating the carcass, not axe #1, and they are using the face that the axe is lying near the carcass to date Axe #1. Since Axe #2 has the same morphology in creation as Axe #1 they are dating Axe#2 to be the same age as Axe#1.