Why does back pain radiate to your limbs?

Nerves: Referred pain is a well known entity. Referred pain is due to nerves much the same is why does a heart attack radiate down your arm. Especially the last 3 lumbar nerves and especially the 1st sacral nerve "go" to your legs/feet. The pain from the back area can be referred down these nerves to make your legs and feet hurt.

Refered pain.: Sensation to your limbs is transmitted by nerves from your back. When there is a pinched nerve in the back, leg may feel numbness, tangling or pain because of this connection. This pain is frequently called sciatica.