Frequency response of amplifier intuition understanding
Shubham Cgawla, do you know where the terms "zero" and "pole" come from? Both terms are related to the complex frequency variable s (which is a theoretical one only). 

For example, the first-order lowpass transfer is H(s)=1/(1+s/wo). This function has a pole (zero denominator) for s=-wo. But note, this is a pure theoretical (artificial) consideration only, because we never can produce a frequency "-wo" (there are no negative frequencies).

Instead, for frequency analyses, we simply set s=jw and find the magnitude of the complex expression [1/(1+jw/wo)]. And - as a result - we get the result that for w=wo the magnitude is 3dB below its maximum (in this case: maximum is "1"). And this statement (cut-off frequency at w=wo) is equivalent to saying "pole at w=wo". 

For a simple RC highpass we can apply similar considerations which lead to a system "zero". 

Summary: The terms "pole" and "zero" are defined for the complex variable s only. But that does not mean that at w=wo the magnitude of such functions becomes zero or infinite in reality. The concept of poles and zeros is only a nice tool to describe the properties of frequency-dependent networks in the complex frequency domain. 

EDIT (Answer to the question in bold):
Quote: The frequency at which an internal capacitor comes out of the open-circuit state is indeed its pole, and its zero is where it enters the short circuit state."

I think, this is a rather "problematic" and unprofessional description. Let´s take - as an example - again the first-order RC lowpass. For very low frequencies the impedance of the capacitor is very large (theoreticcaly infinite for DC). Hence, for some rough calculations it could be approximated as an open circuit. Writing "...comes out of the open-circuit state..." means that the capacitive impedance is in the transition region between "infinite" and finite values which must be taken into account. It can be assumed that the author means the 3dB-corner frequency (which in the complex frequency domain is equivalent to the pole frequency).