Why would stablising selection ever happen?
It occurs when a beneficial characteristic has been developed over time and it would be harmful to stray from it.  In these cases it is not the individuals at the fringe (as you put it the most adapted) who are the best adapted. 

I think it may help if I answer with an example that is widely promoted by AQA in their A2 Biology syllabus. 

Stabilising Selection in human birth weight

It is harmful for an infant to be born with a very low birth weight. They are much more vulnerable to heat loss due to their high surface area to volume ratio and consequently their respiratory demands are very high.  Pre-term babies (which account for 67% of low-birthweight infants(1)) are particularly susceptible to respiratory problems (lack of surfactant in the lungs), cardiac problems (Patent ductus arteriosus - the lungs are still bypassed when the umbilical cord has been cut) and dangerous intestinal problems (Necrotizing enterocolitis) amongst many other conditions can all be fatal (further information on mentioned conditions) and are reflected in high mortality rates at these low birth rates.  It is therefore not beneficial to be on the extremes of birth weight.  

Similarly, delivering a child of too high birth weight can cause complications with delivery if the head and shoulders are too wide to pass through the mother's hips.  Therefore the other extreme of high birth rate is also not beneficial and will not be selected towards.

This leads to selective pressures in both directions, stabilising towards a mean birth weight as shown below:  



This is an example of evolution not pushing a species forward but ensuring that individuals have the best chance of getting to reproductive age themselves.  

(1) Martin, J.A., et al. (2007). Births: Final Data for 2005. National Vital Statistics Reports, 56(6).