The feet contribute to back pain by working their way up the legs. When a foot is pronated (flatter) it causes the lower leg bones to rotate inward, which then causes the thigh bone to rotate inward, which then causes the ilium of the pelvis to rotate forward, which causes the spine on that side to be in more extension than the spine on the other side if that foot is less pronated (which is common).
There are muscular adaptations along the way too, creating weakness, excessive lengthening, or shortness in the muscles controlling these bones. In terms of back pain, the muscles controlling the pelvis and lining the sides of the spine become short making it difficult for the pelvis and spine to return to a normal alignment. Especially if the problems creating the pain are not reversed. Over time, this breaks down other tissues in and around the spine causing pain.
Foot pronation or supination (a higher arch) can be corrected through changing foot strike mechanics. This then can correct these adaptations occurring up the leg and in the spine.