The best (most simplistic) way to explain area rule is to imagine what happens when an aircraft flies through a thin vertical slice of the atmosphere. Upon first reaching the slice its nose pushes the air apart to produce a very small hole. This hole then expands to accommodate the increasing cross section of the fuselage. The rate of expansion suddenly increases as the leading edges of the wings pass through. This is followed by a sudden contraction as the trailing edges pass through, leaving only the rear fuselage. The passing of the fin and tailplane then produce similar, but smaller expansions and contractions.
Expansion of the hole requires the air to accelerate outwards and contraction of the hole requires the air to accelerate inwards. The rate of acceleration is proportional to the rate of change of the cross section of the aircraft. So very large rates of change of cross section require very large rates of acceleration. But the forces necessary to generate these accelerations are provided by the aircraft. This adds to the total drag affecting the aircraft.
For minimum drag it is necessary to minimise the rates of change of the cross section of the aircraft. In the case of a complete aircraft this would involve narrowing the fuselage to offset the sudden increases in cross section caused by the wings, fin and tailplane. The degree of narrowing required in these areas can be reduced by increasing the cross section in the area between the wings and fin/tailplane. It is this narrowing and broadening of the fuselage that gives area ruled aircraft their coke bottle shape. (The term coke bottle shape has been used in JAR POF exams)
The benefits of area rule are greatest for aircraft flying close to the speed of sound. This is because at such speeds, even moderate accelerations will cause the airflow to go supersonic, producing additional drag due to the formation of shockwaves. At very low speeds the extra weight and skin friction can increase the total drag, so (as always) it isn't all good news.