Why is it that, that is "just" what the Zeta function is? What happened in between? I messed around with it for roughly an hour and couldn't get it to come out right.
The second photo is just for the exact definition of how we are taking the Zeta function.
By definition, if $X = \mathbb{P}^1$, so that $N_m = q^m + 1$, then \[ Z(X,s) = \exp\left(\sum_{m = 1}^{\infty} \frac{q^m + 1}{m} q^{-ms}\right) = \exp\left(\sum_{m = 1}^{\infty} \frac{q^m}{m} q^{-ms} + \sum_{m = 1}^{\infty} \frac{1}{m} q^{-ms}\right),\] which is \[\exp\left(\sum_{m = 1}^{\infty} \frac{q^{-m(s - 1)}}{m}\right) \exp\left(\sum_{m = 1}^{\infty} \frac{q^{-ms}}{m}\right).\] As \[\log \frac{1}{1 - z} = \sum_{m = 1}^{\infty} \frac{z^m}{m}\] for $|z| < 1$, it follows by letting $z = q^{-(s - 1)}$ in the first term and $z = q^{-s}$ in the second term that \[Z(X,s) = \exp\left(\log \frac{1}{1 - q^{-(s - 1)}}\right) \exp\left(\log \frac{1}{1 - q^{-s}}\right) = \frac{1}{(1 - q^{-(s - 1)}) (1 - q^{-s})}\] whenever both $|q^{-(s-1)}| < 1$ and $|q^{-s}| < 1$. By analytic continuation, this identity extends to all $s \in \mathbb{C}$ for which the right-hand side is well-defined.