Why does dividing a number with $n$ digits by $n$ $9$'s lead to repeated decimals? For example, $\frac{1563}{9999} = 0.\overline{1563}$.
Why does that make sense from the way the number system works? 
I can vaguely see that since the number $b$ with $n$ $9$'s is always greater than the dividend $a$ of $n$ digits (except in the corner case where they are the same), you will always get $0$ for the units place, after which in the long division process you will be able to divide $a \cdot 10^n$ by $10^n$, $a$ times, which implies you'll be able to divide $a \cdot 10^n$ by $b$, $a$ times with a remainder of $a$, thereby perpetuating this process over and over.
However, I feel this isn't a very precise way to think about it...

This is because  the example can be rewritten as
\begin{align*}\frac{1563}{9999}&=0.1563\cdot\frac 1{1-10^{-4}}=0.1563(1+0.0001+0.00000001+\dots)\\
&=0.1563+0.00001563+0.000000001563+\dots
\end{align*}
