On the dependence between $\bar{z}$ and $z$

Don't blame yourself for not understanding it. It is not clear at all.

It seems to me that whoever wrote that was aiming at the concept of Wirtinger derivatives: if $U\subset\mathbb C$ and $f$ is a map from $U$ into $\mathbb C$, then, if you see $f(z)$ as $f(x+yi)$ (with $x,y\in\mathbb R$, we define$$\frac{\partial f}{\partial z}=\frac12\left(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\right)\text{ and }\frac{\partial f}{\partial\overline z}=\frac12\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right).$$In a sense, $\frac{\partial}{\partial\overline z}$ measures how much $f$ depends on $\overline z$; for instance, $\frac{\partial\overline z}{\partial z}=0$ and $\frac{\partial\overline z}{\partial\overline z}=1$. And $f$ is differentiable at $z_0$ if and only if $\frac{\partial f}{\partial\overline z}(z_0)=0$.