Obviously wires heat up too, but why do they heat up? And for the same reason, why do we get electrical burns?
When electricity moves through anything -- wires or bodily tissues -- there are actual electrons (typically) moving. These electrons are being pulled along by an electric field, but they're also bumping into the atoms that make up the wire or bodily tissue. When an electron bumps into an atom, it transfers some of its kinetic energy to that atom. Temperature is just a form of kinetic energy, where you subtract the "bulk motion". If the atoms in the wire or the bodily tissue aren't moving, this means that all of the kinetic energy that gets transferred to them is measured as temperature. That is, the atoms heat up. This process is called "Joule heating", and is the same principle for wires as for bodily tissue.
The current in the conductor is due to the drifting of electrons inside a conductor in a direction opposite to the flow of electrons During their drifting they collide with their atoms vibrating about their mean position and lose some of kinetic energy to the vibrating atoms which increases the amplitude of the vibration of the atoms thermal energy of the metal with the corresponding rise in temperature of the conductor
All wires which have electric current flowing through them have electrons moving through the wire. The reason for why wires heat up when a current flows through them is that a battery converts chemical energy into electric potential energy. This electric potential energy is given to the electrons, and since the electrons try to minimize their electric potential energy, the electrons convert this electric potential energy into kinetic energy. Due to the wires having electrical resistance, which means that they resist the motion of electrons, the electrons bump into atoms on the outside of the wire, and some of their kinetic energy is given to the atoms as thermal energy. This thermal energy causes the wire to heat up. Electrical burns occur when you hold on to a wire, and due to heat transfer, some of this thermal energy is transferred to you. When the wires get hot enough due to a constant input of heat, there will be more heat to transfer to you if you touch the wire, and all of this excess heat can cause burns and fires.