I know that it works, but I'm seeking a way to think about it so it makes sense.
It seems like a weird question to me. I might not be able to articulate it, and it might not even have an answer (other than it is what it is; it's just defined that way so it works). Let me try to demarcate what I don't understand by saying what I already do understand:
On a number line, for $0<a<b$, the subtraction $b-a$ is the difference (and the distance) between the two points. If you translate both $a$ and $b$ by the same amount left or right, the relative position of the two won't change - even if shifted so far left that $a<0<b$ or $a<b<0$.
A similiar view is "zeroing" at point $a$ (marking point $a$ as "$0$"). The distance from point $0$ to point $b$ is now $b-a$, which we can just read off.
In terms of arithmetic,
for $0<a<b$, their difference $b-a$ is simply subtraction.
For $a<0<b$, the distance between them is the sum of their distances from $0$, or $|a|+|b|$. Since $|a|=-a$ and $|b|=b$, the sum $|b|+|a|=b-a$.
For $a<b<0$, the distance is $|a|-|b|$ (because both magnitudes are reflected, $|a|>|b|$). This is $(-a)-(-b)=-[(-b)-(-a)]=b-a$.
Why does subtraction work out so neatly, so the difference is always $b-a$? What is the property of the rules of arthmetic that make this work? Proofs seem to be about showing $what$ is true, but not $why$. Biology and engineering often have a story about design decisions: what is the underlying design here?
I'm not asking about $|b-a| = |a-b|$; but the simpler, basic subtraction. (To show I understand that equality: by definition $|C|=|-C|$. If $b-a=C$ then $|b-a|=|-(b-a)|$, which is $|a-b|$.)
Background: I've just refreshed my arithmetic by doing all problems in the Khan Academy arithmetic unit (and all material in the negative numbers subsection).
EDIT related question: Subtraction of a negative number