Two factors must be taken into consideration in explaining this periodic trend

(1) Increasing nuclear charge

(2) Increasing shell

Along a period (left to right) the the atomic number increases while the valence electrons remain in the same shell. Thus due to the increasing nuclear charge (pulling electrons closer to the nucleus) the radii of the atoms decrease left to right.

Top to bottom along a group the atomic number continues to increase. However the shell increases from shell 1 to shell 2 etc.. The atomic orbitals for each successive shell get larger and larger - more than compensating for the increased nuclear charge. The result is atomic radii increase top to bottom along a group.