For addition I can see the binary addition at work. The ding "carries" if it get put in a ring that already has a ding by getting its own ring etc... For multiplication I just WAT
1×2 + 1×4 + 0×8 + 1×16, 1×8
and substitute the second for every 1 in the first,
8×2 + 8×4 + 0×8 + 8×16
then finally un-distribute,
(1×8)(1×2 + 1×4 + 0×8 + 1×16)
and that's why putting small copies of the second number in the place of every "ding" in the first number results in multiplication.