From where does the poles and zeros come into the transfer function of any system? And I am not looking for an answer in terms of mathematics but want to know what is there in any physical system that turns out forming poles and zeros when represented in the transfer function form. As in terms of mathematics, we have the dynamics of the physical systems in the form of differential equations which we transform using Laplace transform for the ease of calculation. Then using the poles and zeros, which are the solutions of the denominator and numerator polynomials of the transfer function, we do the stability analysis and rest all mathematical analysis. But this does not give me an intuition of what is there in the physical system that turns out to be a zero/pole in the system transfer function or what physical phenomenon in a system ultimately ends up forming a pole and zero.
At first - the function we call "transfer function" or "system function" is a function of the COMPLEX frequeny variable (s=a+jw) which is just an artifical one. In reality, there are no "complex frequencies". However, it is very convenient to use the complex variable s for describing most of the observations and effects within the frequency domain. Therefore, in most cases there are no real ("physical") poles or zeros.
However, there are some exceptions:
(1) The complex transfer function for some filter circuits (notch, Cauer, Chebyshev invers) has one or more real zeros. That means: The output signal approches zero at the corresponding frequency.
(2) An oscillator producing self-sustained sine wave signals at a frequency fo has a transfer function with a real pole at this frequency fo.
That means: The gain approaches infinity - and a vanishing input signal can produce an output signal. Hence, we do not need any input signal for producing an output. And - because this condition is valid for one single frequency fo only the output will consist of a sine wave fo only. (Remark: This is a - more or less - theoretical consideration, in reality we are not able to place the pole exactly on the imaginary axis, but this is a practical limitation due to tolerances etc.).
Energy storage (resonators) often are the poles (2 at a time).
In feedback systems, the cumulative phaseshifts lead to near-360 degree situations, and these often seem as resonators particularly if lightly dampened.