How and why does sample frequency Fs be mapped to 2π?

Question

Q1: Can anyone give a further explanation on this sentence?

Each time $f_o$ is a multiple of $F_s$, the argument of the exponential is a multiple of $2\pi$

Q2: - must $f_o$ be a multiple of $F_s$?

Answer 1

Your first question:

Can anyone give a further explanation on this sentence? “Each time $f_o$ is a multiple of $F_s$, the argument of the exponential is a multiple of $2\pi$”

This is a simple result based on the provided expression: $$e^{j2\pi \left[\frac{f_o}{F_s}\right]n}$$

Let $f_o = mF_s$, where $m$ is an integer, then: $$e^{j2\pi \left[\frac{f_o}{F_s}\right]n}$$ $$e^{j2\pi \left[\frac{mF_s}{F_s}\right]n}$$ $$e^{j2\pi mn}$$ and now the term in the exponent is equal to $j2\pi$ times an integer $mn$.

Your second question:

And another question - must $f_o$ be a multiple of $F_s$?

No, $f_o$ is unconstrained subject to the answer to your first question.