Why does a metal spoon left on a bowl of hot soup get warm?

Because energy is being conducted to it by conduction

The spoon inside the soup bowl heats up due to heat conduction.

Conduction is one of the types of thermal transfer that allows heat to be transmitted from a hot object to a cold one. (from soup bowl to spoon)

In conduction there is no matter transport. What is transferred is heat, which is a form of energy, corresponding to the increase in the kinetic energy of the atom.

Thermal conduction involves the propagation of heat by contact.