Why does atomic size increase down a group in the periodic table?

Going down a group, two effects compete, and the size-increasing one wins. As you move down, each element adds a new principal shell, so the outermost electrons sit in shells that are farther from the nucleus, which increases the radius. It is true the nuclear charge (number of protons) also rises, which would pull electrons in. However, the inner filled shells shield or screen the outer electrons from this growing nuclear pull. This shielding largely cancels the extra nuclear charge felt by the outermost electrons, so the effective nuclear charge on them stays roughly similar. With the added shell distance dominating and shielding offsetting the charge increase, the net result is that atomic size increases down a group. Across a period there is no new shell, so there the size shrinks instead.