I don't understand this sentence:
By Cauchy-Lipschitz' theorem, every non-constant solution of $y'(t)=f(y(t))$ is necessarily monotonic because the codomain of its values has to be contained by an interval in which f is constantly positive or negative.
First of all: is it true? And could anyone explain it to me? (Why is it true? or why is it not)
It is true, whenever the Cauchy-Lipschitz theorem is applicable.
To see this, assume the contrary that $f(y(t))$ is not always positive or negative. Then there is $s$ so that $f(y(s)) = y'(s) = 0$. Hence $y(t)$ and the constant function $y_1(t) = y(s)$ both solve the IVP
$$\begin{cases} x'(t) = f(x(t)) & \\x(s) = y(s).\end{cases}$$
which contradicts the uniqueness theorem.