Why does $z^n-1=0$ have at max n solutions? $z\in\mathbb{C}$ I know that there is a Theorem which says that a Polynom of Degree n has at most n Solutions, however we have not proved it yet in our class. Is there Maybe another explaination for this Special case?

There's an explanation if you represent them in polar co-ordinates and consider that multiplying two complex numbers involves adding their arguments (ie angles) and multiplying their magnitudes (distances from the origin). It turns out they need to have $1$ as their magnitude and be multiples of$\frac{360°}{n}$ apart on the resulting circle, so only $n$ of them will fit.
(This is effectively the same as gimusi's answer.)
Edit: In fact, there are exactly $n$ of them, and they're equally spaced round the circle. The reason should be obvious if you pick one of the candidate angles and multiply it by $n$.
