The center of gravity (where you could consider all of the mass to be) is right above your finger.
If you hang a mass off one end of the stick, the center of gravity of the system moves toward that end. Thus, you have to move your finger in that direction to keep it balanced.
Now we are talking about two forces (or since your finger acts as a pivot point, two torques).
By observing how far the pivot point, or fulcrum moves, we can find the mass of the meterstick.
I love this activity because students percent error is almost always under 1%.
