Here, a "neutral atom" is simply an atom that has no charge.
See, an atom consists of protons, neutrons, and electrons. Protons are positively charged, electrons are negatively charged (with the same magnitude of charge per particle as a proton). Neutrons have no charge.
Now, in a "neutral atom", the number of protons must be equal to the number of electrons, otherwise it would not be neutral.
Basically, the definition is saying that "in a neutral atom, the atomic number is equal to both the number of protons , and the number of electrons, because, well, they both are the same."
Compare this with an ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge
Here, the atomic number cannot be equal to both--because they are unequal. In this case, a better definition would be:
The number of protons present in an atom is it's atomic number. It is represented by $Z$.