I can only answer on a cellular level: the skin cells of the octopus/squid have little packets of pigment stored within the cells. When the squid wants to change color, its brain sends a signal to these cells, which directs some proteins to transport these packets from the interior of the cell to the exterior (cell plasma membrane).
The packets fuse with the membrane and the pigments are exposed to the outside. As cells across the skin perform the process together, the squid appears to change color.
To change back, the process is reversed and the pigments are pinched back into their packets and stored away to be reused.
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