my understanding is that there is a thin trace (a fuse) at each cell. in order to program it, the address lines are manipulated to select the word, and a programming voltage is applied (for some reason I remember 18v, but it really could be anything), that draws a large current across the fused and blows it.

on a die, you wouldn't go through that trouble, you would just directly synthesize the zeros and ones.

but we don't really use ROMs that much anymore, they are usually persistent and programmable