The handshake began as proof you weren't hiding a weapon

The handshake is thought to originate in ancient Greece and Rome as a gesture of peace — extending the right (weapon) hand open to show it held nothing. Medieval knights would also shake hands to dislodge any concealed weapons up a sleeve. It became a symbol of trust precisely because it began as a trust-verification mechanism.

Source: Westland, Pamela, 'The Language of the Body' / Classical history records