The other replies are simplifying too much. Just adding or subtracting in a loop would be far too slow.

A multiplier will find the partial products by using AND gates, and then sum them, which is very similar to long multiplication as they teach you in school. This article explains it pretty well.

Division is more complicated. It's sort of done like long division, but apparently that is slow and there's some magic with two's complements that can make it faster. Honestly I don't fully understand it yet.