Why $f(x)=x^e$ is a bijection i.f.f $e\in{\mathbf{Z^*_{\phi(N)}}}$? I understand that if $e\in{\mathbf{Z^*_{\phi(N)}}}$ then $\gcd(e,\phi(N))=1$
and if $e\not\in{\mathbf{Z^*_{\phi(N)}}}$ than $\gcd(e,\phi(N))\neq{}1$.

But I couldn't figure out why this implies bijection of $f(x)=x^e$.

I also tried to see examples but that didn't help me to explain the phenomenon.
The correct form is:
The integer $a \mod Z_n$ has multiplicative inverse iff $gcd(n,a)=1$

Here, you are working on exponents, so you must consider the modulo as $\phi(n)$ not $n$.

Therefore, here you can find the inverse of $e$ iff $gcd(e, \phi(e))=1$. This guarantees that you be able to find the inverse of $e$ using extended Euclidean algorithm.