No, that's not ROM. It's nonvolatile memory. A 1 or 0 is stored in each core depending on which way the core is magnetized. To store a bit, the cores are on an X/Y grid. You'd send a current pulse 1/2 that required to flip the state on the appropriate X and Y lines. Only the core at the intersection would see enough current to be set. To read it out, you would do the same thing and a sense wire would have an induced current from the magnetic field change. A bit simplified, but that's it in a nutshell. It's been a long time, but I think reading is destructive; you have to write the bit back after reading it.
There's a lot of support circuitry required: x and y drivers, sense amps plus logic to do something with it.