How could a common cold be spread by shaking hands with someone?

When you have a cold, the viruses in your body and on your skin will infect your cells and start to multiply rapidly, as viruses do inside the host. Now imagine all those virions (virus particles) on your hands (which are nearly impossible to avoid without washing them first) and you shake someone's hand. The viruses, that are not actually living organisms -- just "chemicals", can be rubbed onto the hand of the other person when you shake or hold hands. All the other person needs to do to become infected, too, is to rub their nose, mouth, or eyes which moves the virions from their own hand to where they can get inside the mucous tissue cells, and they will come down with a cold, too.

Washing your hands physically rubs and washes off the virus. The soap and water loosen the particles so they can be rinsed down the drain. See the related questions below for more information about spreading a cold.