As you move down a group, the outermost electrons are in a higher energy level, so subject to more electric shielding. Therefore, atomic radius increases down a group.
As you move across a period, the number of protons in the nucleus increases, yet the number of energy levels stays the same. Therefore, there is more effective charge on each of the outermost electrons, so the radius decreases. As we are adding more electrons to the same orbital, you would expect there to be an increase in electric repulsion between the electrons. This is true but has negligible effect compared to the increase in nuclear charge.