Effective nuclear charge on an inner electron
Firstly, effective nuclear charge is just a model for predicting trends in atomic properties. The actual solution of the quantum mechanical equations does use the full nuclear charge and explicit calculation of the all electron-electron repulsions (and electron-nuclear attraction).

The point of $Z_\text{eff}$ is to approximate the value of the nuclear charge felt by electrons at a given distance. The answer to your question then is Gauss's Law,  which states that, to determine the field at a given point, you consider all and only the charge inside a certain surface (on which your point sits). Moreover, it states that the field there is equivalent to if all of the charge inside that surface was a point charge at the center.

So if we apply Gauss's Law to our atom, we add up all the charge inside a sphere centered on the nucleus ($Z-\text{effective number of core electrons}$), and regard the system as an effective nucleus with that charge, no core electrons and the electron of interest being the "first" electron and so we ignore the outer electrons.

Here are some physics discussions for why, intuitively, Gauss's Law allows you to ignore outside electrons from the physics stackexchange.