I have a question regarding the poles and zeros at infinity I often read here in DSP SE and also in some textbooks about poles and zeros at infinity
This question also answers somehow (but not in much easy/simple way) Pole/Zero existence at infinity
I am confused about how to realize the poles and zeros at infinity especially from observing/looking at system transfer function??
It's actually quite straightforward: positive powers of $s$ (or, in discrete-time, $z$), correspond to poles at infinity. Negative powers give you zeros at infinity.
Let's look at some examples. In continuous time, an ideal differentiator has the transfer function
$$H(s)=s\tag{1}$$
Clearly, $\lim_{s\to\infty}H(s)=\infty$, hence you have a pole at infinity (and a zero at $s=0$).
The transfer function of an ideal integrator is
$$H(s)=\frac{1}{s}\tag{2}$$
Here we have $\lim_{s\to\infty}H(s)=0$, i.e., a zero at infinity (and a pole at $s=0$).
In discrete-time, a delay of one sample corresponds to
$$H(z)=z^{-1}\tag{3}$$
and you get $\lim_{z\to\infty}H(z)=0$.
A (non-causal) advance by one sample is represented by the transfer function
$$H(z)=z\tag{4}$$
In this case you have a pole at infinity: $\lim_{z\to\infty}H(z)=\infty$
Note that in continuous time as well as in discrete time, systems with poles at infinity cannot be causal and stable.
