More particles in the same space
Concentration is the number of reactant particles in unit volume. So the explanation is a direct one.
As the concentration of a reactant increases, the number of its particles in unit volume increases. So the number of collisions between reactant particles in unit time increases as well. That is why the rate of reaction rises as the concentration of the reactants rises.
A dilute solution
A few acid molecules among a great many water molecules. What the magnesium surface usually meets is water.
A concentrated solution
Many acid molecules in that same volume. Any given spot on the magnesium surface is far more likely to meet one.
A common mistake
Raising the concentration raises the speed at which the reaction goes, not the amount of product obtained at the end. That amount is settled by how much reactant there is.