Why does the Denominator of the Denominator go to the Numerator?

There are two confusions here. One is that the expression 1/2/3, when there are no parentheses, is defined to be (1/2)/3, which is different from the expression 1/(2/3), and that is why you got the wrong answer in your post. The second is why "the denominator of the denominator goes to the numerator", and it is not something you need to memorize, instead you can do the following

$$\dfrac1{\left(\frac23\right)} = \dfrac1{\left(\frac23\right)}\cdot 1 = \dfrac1{\left(\frac23\right)}\cdot \dfrac33 = \dfrac{1\cdot 3}{\left(\frac23\right)\cdot 3} = \dfrac{3}{\frac{2}3\cdot 3} = \dfrac32.$$

That is, if the denominator in the denominator is 3, just multiply the entire fraction by $\frac33$, and this will cancel the denominator in the denominator.

Let me try to explain the formula $$ \frac {\frac ab}{\frac cd}=\frac ab \cdot\frac dc\:\:\:\:\:(*)$$ without too much abstraction. I intend to use only the definition of multiplication of fractions and the rules of solving equations that you study in the school. Let us denote the left side of the previous equality by $x.$ Then $$x=\frac {\frac ab}{\frac cd}.$$ Multiplying both sides of this equality by ${\frac cd}$ we obtain $${\frac cd}x= {\frac ab}.$$ Now multiplying both sides of the equation by ${\frac dc}$ we get $${\frac dc\frac cd}x= {\frac dc\frac ab}.$$ Which is $${\frac {dc}{cd}}x= {\frac ab\frac dc}.$$ Or, equivalently, $$ x= {\frac ab\frac dc}.$$ This proves that the formula $(*)$ must be true.

You need to be careful with the order of operations:

$$ 1/2/2 = (1/2)/2 = 1/4 $$ while $$ 1/(2/2) = 2/2 = 1 $$