The handshake originated as a gesture to show you weren't holding a weapon

The handshake is believed to have originated in ancient Mediterranean societies as a gesture of peace — extending the right (weapon) hand open demonstrated no sword or dagger was concealed. Greek artwork from the 5th century BC depicts handshakes between gods and mortals to seal agreements. The Quakers popularised the handshake in 17th-century England as an egalitarian substitute for bowing or hat-doffing, which they refused to perform on hierarchical grounds. In most Western cultures today, a limp handshake is associated with low confidence or untrustworthiness — an enduring vestige of its origins as a trust signal.

Source: Eibl-Eibesfeldt, Human Ethology (1989)