This is describing the shape of an area in polar coordinates. Polar coordinates work from a known starting point, and each new point is derived from a bearing (angle) and a distance. Think of it this way: you are at your kitchen table. If you face North and turn clockwise 40 degrees, and then walk 5 steps, you're at the hallway door. So your hallway door is 40 degrees and 3.5 meters (considering 70cm per step) from your kitchen table.
EDIT (in accordance to comment by mkennedy): in the USA, bearing is measured either from the North or South, and you turn X degrees to either East or West. So a "North 40 degrees East" bearing means looking North, turn 40 degrees to the East.
Your area description is marking a line, and saying your area is everything encompassing 12.5 ft to the left and right of your centre line. The starting point is "center of section 4, township 23, north range 32 west, McDonald County, MO". From there, you look East, turn 68°39'25" anticlockwise and walk 71.96ft. This is your second point. From it, you look East again, turn 75°51'17" clockwise and walk 55.41ft. That is your third point. Finally you look East again, turn 81°40'50" clockwise and walk 162.27ft to what is your final point. Everything within 12.5ft of this line is your area of interest.