The parametrizations $$z-i = r_1\exp(i\theta_1) \quad \text{and}\quad z+i = r_2\exp(i\theta_2)$$
are used to show how $f(z) = (z^2+1)^{1/2}$ changes when we make a complete loop around the branch points $z=i$ and $z=-i$ respectively.
How can we assume that the curve parametrized by $$(z-i)(z+i)=r_1 r_2 \exp(i(\theta_1+\theta_2))$$ traversed such that $\theta_1 \to \theta_1 + 2\pi$ still constitutes a loop around $z=i$, considering the fact that $$ z-i = r_1 \exp(i\theta_1) $$ is completely different from $$ (z-i)(z+i)=r_1 r_2 \exp(i(\theta_1+\theta_2)) $$?
Furthermore, what observation precludes that $\theta_1 \to \theta_1 +2\pi$ also encloses the other branch point?
Write $f(z)= \big( g(\tilde{z}) \big)^{1/2}$
We can construct a parametrization which simultaneously parametrizes both loops around the branch points $z=i$ and $z=-i$ respectively, by defining $$g(\tilde{z} ):= g(r_1,r_2,\theta_1,\theta_2) = r_1r_2\exp(i\theta_1 + i\theta_2)$$ after which we impose two characterizing constraints:
\begin{align} z &= i+r_1\exp(i\theta_1) \quad \text{($z$ w.r.t. $z=i$)} \\ &= -i+r_2\exp(i\theta_2) \quad \text{($z$ w.r.t. $z=-i$)} \end{align} and $$ g(z) = (z-i)(z+i) = r_1 r_2 \exp(i\theta_1 + i\theta_2) = g(\tilde{z}) \ . $$ By using this magically clever parametrization of the text, we can arbitrarily fix $\tilde{z}$ by varying any of the variables $(r_1,r_2,\theta_1,\theta_2)$, keeping in mind that we have imposed two characterizing constraints. Hence we still have two degrees of freedom; not four.
Therefore, a complete loop around $z=i$ is now simply given by $\theta_1 \to \theta_1 + 2\pi$. Similarly, a loop around $z=-i$ is given by $\theta_2 \to \theta_2 + 2\pi$; the imposed constraints take care of the rest.