Atoms of elements at the top of a group on the periodic table are smaller than the atoms of elements at the bottom of the group. how does this explain the difference in the reactivity of metals within a group?

Explanation:

As we move down the group atomic radii increased with increase of atomic number. The addition of electron in next level cause the atomic radii to increased. The hold of nucleus on valance shell become weaker because of shielding of electrons thus size of atom increased.

As the size of atom increases the ionization energy from top to bottom also  decreases because it becomes easier to remove the electron because of less nuclear attraction and as more electrons are added the outer electrons becomes more shielded and away from nucleus.  

On left side of periodic table atoms of metals are more reactive by loosing the electrons or we can say metals are more reactive by loosing the electrons so their reactivity increase down the group because of easily  removal of electrons.

On right side of periodic table atoms of nonmetals are more reactive by gaining the electrons. As we move down the group nuclear attraction becomes smaller because of shielding thus electron are less attracted by nucleus and reactivity decreases.