How do afferent mechanoreceptors work on the finger pads?
Short answer
Tactile grating stimuli with a higher line density (number of lines per surface area) and applied with an equal force result in a lower amount of pressure per unit of skin area under the grating, and hence smaller afferent responses.

Background
In the Discussion section of the linked article (Phillips et al, 1993) the authors explain on p. 838 (left column) that the reduction in the response of SA afferents (slowly adapting fibers) when grating density is increased is due to the higher proximity of neighboring areas of stimulation, i.e., the lines in the grating stimuli are more closely spaced. The authors reason that this leads to a reduction in the effective stimulus (i.e., compressive strain) at the receptor terminal due to the presence of adjacent stimulus elements that distributes, and hence reduces the skin loads per line. 

Reference
- Phillips et al, J Neurosci; 12(3): 927-39