Why it works: Nikhilaṃ Navataścaramaṃ Daśataḥ

For numbers near base B: let $a = B-x$, $b = B-y$ (deficiencies). Then $a\times b = (B-x)(B-y) = B^2-B$($x+y$)+xy = B($a-y$)+xy = B($b-x$)+xy. The left part is one number minus the other's deficiency; the right part is the product of the deficiencies.