Why does casting out nines work?

Because a number and its digit sum leave the same remainder when divided by 9 — the digital root is just the number mod 9.

Since 10 ≡ 1 (mod 9), every digit contributes only itself to the remainder, so any number ≡ its digit sum (mod 9). Arithmetic is preserved under "take the remainder mod 9": if a·b = c, then (a mod 9)·(b mod 9) ≡ c (mod 9). Casting out nines is exactly checking that congruence. (Digital roots run 1–9; a multiple of 9 gives 9, standing in for remainder 0.)