Since you already have a proof, my best guess is that you're seeking intuition. I can offer a somewhat more intuitive explanation of why this is the case for two values. During the explanation, it helps to think of a concrete example -- let's say the values are 1 and 5.
With two values, the mean is the midpoint of the two values. Both values are equidistant from the mean. (3 is the mean, both values are 2 away from that.)
range is twice the distance of the midpoint to either of the observations. (The range is 5-1=4, and the distance of the ends to the midpoint is half of that -- i.e. 2.)
since both points are equally far from the mean, the mean deviation is that distance of either point from the mean (i.e. half the range). With two points the distance to half-way is always half the distance to the other end.
with standard deviation*, $s_n$, first look at the variance, $s_n^2$. The variance is the average of the squared distances from the mean. Note that each point is the mean deviation away (half the range) from the mean, so when you square those you get both points contribute $\text{MD}^2$; the average of that is also $\text{MD}^2$. When you take square roots to get back to the standard deviation, you will get the mean deviation. ($(5-3)^2 = 2^2, (1-3)^2=2^2$, both of which are just the square of the distance from the middle to the ends; the average of $2^2$ and $2^2$ is $2^2$, the square of the mean deviation)
* this is the $n$-denominator standard deviation, not the Bessel-corrected version.