What do you think the original function of shaking hands was as compared to what it is today?

There was this funny study that found that people were more likely to hold their hand next to their nose right after shaking hands. From this they concluded that hand-shaking is a non-intrusive way of smelling a person you just met.

showing that you're not holding a weapon.

...in one hand.

It also shows that you trust the person not to use a weapon. One hand may be clear but you trust that the person doesn't have one in the other.