How exactly does $\frac{\partial f}{\partial \bar{z}}$ work?

$\newcommand{\dd}{\partial}$The complex differential operators may be defined as complex linear combinations of ordinary partial derivatives: $$ \frac{\dd}{\dd z} = \frac{1}{2}\left(\frac{\dd}{\dd x} - i\frac{\dd}{\dd y}\right),\qquad \frac{\dd}{\dd \bar{z}} = \frac{1}{2}\left(\frac{\dd}{\dd x} + i\frac{\dd}{\dd y}\right). \tag{1} $$ This is what the chain rule would give if you treated $z = x + iy$ and $\bar{z} = x - iy$ as independent (so that $x = \frac{1}{2}(z + \bar{z})$ and $y = \frac{1}{2i}(z - \bar{z})$).

Then it's just a matter of formalities to check that $$ \frac{\dd z}{\dd z} = \frac{\dd \bar{z}}{\dd \bar{z}} = 1,\qquad \frac{\dd \bar{z}}{\dd z} = \frac{\dd z}{\dd \bar{z}} = 0 $$ and $$ \frac{\dd}{\dd z} f(z, \bar{z}) = D_{1}f(z, \bar{z}),\qquad \frac{\dd}{\dd \bar{z}} f(z, \bar{z}) = D_{2}f(z, \bar{z}), $$ i.e., that a function expressed in terms of $z$ and $\bar{z}$ can be (formally) differentiated using the standard manipulations as if $z$ and $\bar{z}$ were independent variables.

If you must know, there's a rigorous construction (well-known to all students of complex geometry) that involves complexifying the tangent bundle of $\mathbf{R}^{2}$—equipped with the "complex structure" $J$ that rotates each tangent plane a quarter turn—and splitting the complexified tangent bundle into eigenbundles of (the natural extension of) $J$. The formulas (1) then define a complex-linear isomorphism between $(T\mathbf{R}^{2}, J)$ and the $i$-eigenbundle of $J$, and an anti-linear isomorphism with the $(-i)$-eigenbundle.