As we move from left to right in a period, the number of shells of the atom of every element remains same, but the number of electrons in the outermost shell and the number of protons goes on increasing. This causes an increases in the nuclear force, which brings the elctrons slightly closer to the nucleus, thereby reducing the atomic radius and size of the atom.
Now when we reach the noble gases, their atoms contain 2 (Helium) or 8 (all others) electrons in the outermost shell. Due to such a high amount of electrons in the outermost shell, the electrons develop a repulsive force among each other. This repulsive force overcomes the increased nuclear force, causing the electrons to move away from each other, effective increasing the atomic radius and atomic size. Hence, due to the force of repulsion between the valence electrons, the atomic size of atoms of noble gases is slightly greater than the size of the atoms of the elements immediately left of it in the periodic table.
Because there is a high level of electrons on the outer level, the amount of electrons on the outer level causes the electrons to repulse each other and move father away from each other, increasing the size.