The atomic radii of atoms increase as you travel down a family on the periodic table because of the increased number of electron shells. Each additional shell adds a layer that disrupts the attraction between the protons in the nucleus and the electrons in the shells that pull an atom together.
Elements in a column or family on the periodic table of elements contain similar properties, including the exact same total number of valence electrons, which are the electrons found in the outermost electron shell of an atom. The negatively charged valence electrons are the most reactive and easily attracted electrons to both the positively charged protons in the nucleus and external forces of attraction.
The greater the number of filled electron shells between the protons and the valence electrons, the weaker the forces of attraction experienced between them. The effective nuclear charge of the positive protons in turn increases the atomic radius and overall size of an atom. Because the number of filled electron shells increases when traveling from top to bottom of a family of elements in the periodic table, the atomic radii experience a decreasing trend. This phenomenon ties into other properties, such as electronegativity, which experiences a similar trend across periodic families and is determined by the same forces.