Why Electric current is the flow of charge, and in a solid that means electrons moving. Metals have atoms whose outer valence electrons are only loosely bound, so they drift freely through the crystal lattice as a 'sea' of free electrons. Apply even a small voltage and those electrons move easily, which is exactly what low resistance means. Insulators, by contrast, hold their electrons tightly in place.
Watch out Protons sit locked in the atomic nuclei and never move through a wire, so free protons are not a thing in metal conduction. Density and Young's modulus (stiffness) are mechanical properties with no direct bearing on electrical conductivity, though lead is dense and a poor conductor, which shows they do not track together.
Current is electrons moving. Metals are good conductors because their electrons are free to roam.