A qualitative way to understand this trend is that with each increase in n, going row by row down a column, the electrons of filled inner shells create a screening effect upon the charge of the nucleus. The nuclear charge grows with each added proton, but as electrons are placed into new shells (with larger n) above previous shells, they experience not only the attraction of the nucleus but the repulsion of the inner electrons.
This doesn't explain the increase in Z_eff down a column by itself though. For that, we also need to consider that the radial probability distribution (The likelihood of finding the electron a certain radial distance from the nucleus) of the (n)s, (n)p, (n-1)d, and (n-2)f electrons within a row, changes to reflect the appearance of nodes as n, the energy level, increases.
Consider the s orbitals and the alkali metals/hydrogen. The 1s orbital (Hydrogen) has no radial nodes in the probability distribution, while the 2s (Lithium) and 3s (Sodium) have one and two radial nodes, respectively. The nodes divide up the radial probability into peaks (Technically, I think it's like n - l peaks, but you asked for a qualitative explanation). As n gets larger, the most probable peak moves farther away from the nucleus (Which should hopefully make sense), but the other peaks appear deeper within the atom, closer to the nucleus. These smaller peaks of probability can actually appear deeper (Shorter radial distance) than some inner "Shielding" orbitals and allow the electron described by that probability distribution to feel an effectively less screened nuclear charge. This happens even though this electron is more probable to be found farther away than the shielding layers of electrons underneath it.
I found this image from the cited online textbook to be illustrative of this example I am talking about:
I am referencing this textbook's section on "Electron Penetration": https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/07._Periodic_Properties_of_the_Elements/7.2%3A_Shielding_and_Effective_Nuclear_Charge