In your example (a hole in a bar loaded axially), the hole stretches. This causes the stress around the hole to vary. The locations where the hole changes curvature the most is where the stress is a maximum. From this, the stress concentration can be calculated: k=max stress/average stress.
Here is an image from a finite element analysis (FEA) showing the elongation and stress around the hole. Red is high stress, dark blue is low stress. Note that the minimum stress is lower than the average stress.
Edit: The hole in the original bar is circular, and the above image greatly exaggerates the displacement. Also, you need to check the reference for the stress concentration factor K to see if the average stress = force/w/t (the total area) or average stress = force/(w-d)/t (the net area), where w is the width of the bar, d is the diameter of the hole, and t is the thickness of the bar.