A metal wire is made of a lattice of positive ions, with free electrons moving between them. When there is a current, the electrons drift through the metal.
As the electrons move, they collide with the ions in the lattice. In each collision some energy is transferred from the electrons to the ions. The ions vibrate more, so the metal gets hotter. This is the cause of the heating effect and of electrical resistance.
A higher resistance means more collisions, so more energy is transferred to the lattice. A bigger current means more electrons pass each second, so there are also more collisions each second and more heating.
