Why do the atomic radii of the elements decrease across a period?

Moving across a period the number of protons and neutrons that make up the nucleus of the element increases. The increase in the number of positively charged protons results in an increased nuclear charge centred at the nucleus. This has the effect of increasing the amount of attraction to the electrons around the outside of the element. Subsequently, the electrons are pulled closer to the centre of the element and the overall size is reduced.