As you move from left to right across a period, the number of protons in the nucleus increases. The electrons are thus attracted to the nucleus more strongly, and the atomic radius is smaller (this attraction is much stronger than the relatively weak repulsion between electrons).
As you move down a column, there are more protons, but there are also more complete energy levels below the valence electrons. These lower energy levels shield the valence electrons from the attractive effects of the atom's nucleus, so the atomic radius gets larger.
You might wanna trim that down a little, but you got the gist of it.