Just why this occurs is made clear when we understand what vitamin B6 usually does. Under normal conditions, the vitamin acts as a cofactor in the synthesis of several neurotransmitters: adrenaline, noradrenaline, serotonin and levodopa to be precise. Clearly, if the production of neurotransmitters is impaired, it isn't that difficult to see why a dysfunction of nervous tissue might result. The reason the peripheral nerves are the most frequently involved is simply because they are the longest, and thus the susceptible to any perturbation. (In almost all cases of peripheral neuropathy, the damage to nerves in the body is pretty general, and it is only because the peripheral ones are the biggest targets that they are most affected.)
Although a dietary deficiency of vitamin B6 is fairly uncommon in the over-nourished Western world, symptoms of this vitamin's deficiency come to the fore when using certain drugs. In particular, isoniazid (a first-line anti-tuberculous drug) not uncommonly causes a peripheral neuropathy, and it does this by competing with pyridoxine in its action as a cofactor of those neurotransmitters' synthesis. To counter this possibility, patients treated with the drug are often supplemented with pyridoxine, especially if they have other risk factors for peripheral neuropathies, such as diabetes and HIV.