When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of times)? In other words, why does 1 keep appearing? I'm looking for the visual intuition.
Here's my attempt to answer this:
Start with this definition: Division is "equal distribution."
Consider $\frac{2}{3}$ as equally distributable, as part of $\frac{1}{3}\times{3}$.
Realize that 1 frequently appears because (probably) it demonstrates #1. If you divide something into 2 equal parts, each part is $\frac{1}{2}$ -- that is, its multiplicative inverse. So $\frac{1}{2}$ $\times$ $\frac{2}{1}$ = 1 which indicates that you've equally distributed.
Here is a sketch:
Update: This works because $\frac a b = c$ (where a,b, and c are rational numbers and b is not zero) can be rewritten as $a = b \times c$
Well, your pictures are not correct. I recommend you to watch these Khan Academy videos:
Edit: Or I am not able to see what you have meant by those pictures.