Zeff is the effective nuclear charge which is the attractive force felt by an electron from the positively charged nucleus. With the exception of H which only has one proton and one electron, all other atoms will experience the effects of shielding.
There will be a certain attractive force between the nucleus (i.e. protons) and the electrons. However, shielding will occur because the presence of other electrons will result in the "blocking" of some of that attractive force. Electrons in lower energy levels than the electron we are looking at provide a lot of shielding. Electrons in the same energy level oribitals will provide only a small amount of shielding.
As we go across a period, the amount of attractive force (proton-electron interactions) increases. The amount of shielding also increases because we are adding electrons to orbitals at the same energy level. However, the attractive force between the protons and electrons increases more than the shielding so we see a reduction in the atomic radii.
As we go down a group, the amount of attractive force between the protons and electrons still increases, but now the increase is "overpowered" by the large increases in shielding that result from adding an electron to a higher energy orbital that is feeling the full shielding from all the electrons in lower energy orbitals. As a result, the atoms get larger as we go down the column.
Zeff changes more/increases as you move through a period because as the Z (number of protons) increases the effect of shielding on the valence electrons and their distance from the nucleus remains constant.