The wires running inside of the ferrite beads create the fields that flip the direction of magnetization, writing a zero or a one. They also sense the magnetic field so the computer can read the stored data. Two of the wires organize the core into an X-Y matrix. The core plane is an array of vertical and horizontal wires with a bead at each intersecting node. As the bit changes state, it induces a positive or negative pulse in a third wire that runs through all of the cores in a plane.
Sensitive amplifiers convert the positive or negative signal to a corresponding zero or one.
Since the amplifiers detect nothing unless the core changes state, reads are destructive. You've got to toggle the bit, and then write the data back in, on each and every read cycle. It sounds terribly primitive till one thinks about the awful things we do to keep modern DRAM alive. Before microprocessors quite caught on, the instrumentation company where I worked embedded Data General Nova minicomputers into products. The Nova used core arranged in a 32k x 16 array.