The short answer is that it's a unit conversion factor. Physicists often work in units where c = 1, and then the equation becomes E=m multiplied by a unit conversion term.
If you imagine (rest) mass to be a special form of energy to begin with, you're already 99% there. The fact that the speed of light squared is used as a conversion factor is superficially due to the way our units system is constructed, and in a deeper way it is due to the fact that fundamental properties of the universe are reflected in all of its fundamental processes (so, in a way these are also conversion factors in the actual mechanics of the universe).