Why do atoms get smaller moving left to right on a periodic table? I get that it's because of the increase number of protons, but I still don't understand the concept. When you move left to right on a periodic table, the protons increase at the same rate as the electrons. In fact,  the number of electrons end up being the same as the number of protons, no matter how far you are across a period (unless it's an ion).  

So shouldn't the attractive force of the nucleus and the forces of electron-electron repulsion be cancelling out due to the equal number of protons and electrons in an atom? Or is it because protons have a greater force of attraction compared to the repulsion force between electrons?
Atomic size across a period, in fact, has everything to do with electron shielding and the effective nuclear charge (charge felt by the electrons).

Across the period, electrons are added to the same outer level, meaning that shielding by inner electrons does not change, however, the numbers of proton increases. This results in the effective nuclear charge rising significantly (as no. protons increases while added electrons are shielding inefficiently), which of course, means that the atom has a smaller size.