What actually happens when your limbs "fall asleep"?

Nerve conmpression : Typically you have compressed a proximal nerve root. If limb is "cold" and "numb" you could compress nerve and blood supply (typically they run together). If the pressure is relieved, feeling comes back. You can try this yourself at your wrist level by squeezing tight around wrist or putting watch too tight. Just make sure to let go before hand turns completely white!