Logic behind dividing negative numbers

If you have a debt of \$100, and that debt is paid in \$20 increments, you have:

$$\frac{-\$ 100}{-\$20} = 5\ \textrm{payments}.$$

Do not attempt to base your understanding of negative numbers on analogies with debts or with visualization. The sooner you grasp the concept of numbers as abstract objects, the better off you'll be.

The division $\frac a b$ means "what number, when multiplied by $b$, gives $a$?". Because of the way negative number multiplication is defined (and it is a made up, abstract definition, it doesn't "come" from anywhere), it is the case that multiplying a negative number $-b$ by the positive number $\frac a b$ gives you $-a$. Therefore, the number that needs to be multiplied by $-b$ to give $-a$ is the positive number $\frac a b$.

EDIT:

To respond to criticism, I'd like to clarify what my point of view with an example. Most people think of positive numbers as "abstract objects", in the sense I'm using here. "$5$" isn't "$5$ meters", it's just $5$. Of course, we're aware that they have concrete "representations", but the abstract point of view actually helps us there, because it gives us a bird's eye intuition on which things can be represented as numbers and which can't (anything that can be counted, basically). Now look at the ancient Greeks and their successors. They forced themselves to represent numbers as lengths. Multiplication of two numbers meant considering the area of a rectangle. Multiplciation of three meant considering the volume of a cuboid. This led to mathematicians saying patently ridiculous things like that multiplying four numbers together is illogical (specifically, that quartic equations are absurd, something that I think Cardano said).

Interpret multiplications as a way of scaling.

$2\times3$ would be the length $"2"$ scaled up thrice.

This provides a nice way to interpret negative multiplication as scaling and reversing.

$9$ meters forward times $-2$ would be $18$ meters backward.

Now define division as the inverse of multiplication. In other words, $-18/-2$ is defined to be the solution to the equation

$$(-2)x=-18$$

Now ask yourself, what length, should I double and flip to get $18$ backwards?

Clearly the answer is $9$ forwards or $+9$.

Note: The other answer here already has an interpretation with debts like you were asking for, I just posted it because the scaling thing is a nice way to look at it. (It also extends nicely to the complex numbers, showing you why they are not-so-imaginary)