In fluid dynamics and elasticity theory (and probably many other theories Im not familiar with) , when we consider a 2D "flat" case, we summon complex analysis for help. It usually starts with introducing some potentials, partial derivatives of which are equal to something we are interested in.
Even though I've completed a course of complex analysis, to me all of this seems like magic. I don't have any intuition or know any intrinsic reasons for why this helps us solve "flat" problems.
What should I read up on, or draw my attention to, in order to gain some intuition for why this sort of approach is used, why it is needed, and why it works?
The reason is that many physical problems boil down to solving the Laplace equation $\Delta\varphi=0$ on some domain $D\subseteq\mathbb R^2$ with given boundary conditions $\varphi(z)=\varphi_0(z)$ for all $z\in\partial D$. And as it turns out, $\varphi$ is a solution to the Laplace equation iff it is the real part of some holomorphic function on $D$ (viewed as a subset of $\mathbb C$). In addition, the composition of any two holomorphic functions is holomorphic as well. So if $f:D\to E$ is a bijective holomorphic function (also called conformal map) between the domains $D$ and $E$, and $\tilde\varphi$ satisfies the Laplace equation on $E$, then the holomorphic function $\Phi:E\to \mathbb C$ whose real part is $\varphi$ can be composed with $f$ to get a holomorphic function $\Phi:=\tilde\Phi\circ f$ whose real part $\varphi$ satisfies the Laplace equation on $D$. And if $\tilde\varphi$ satisfies the boundary conditions $\tilde\varphi(w)=\varphi_0\circ f^{-1}(w)$ for all $w\in\partial E$, then $\varphi$ also satisfies the boundary condition $\varphi(z)=\varphi_0(z)$ for all $z\in\partial D$. So if we solve the Laplace equation on $E$ with transformed boundary conditions and then transform back to $D$ using a conformal map, we have solved the original Laplace equation with the original boundary conditions.
So if we have a domain $E$ on which the Laplace equation is comparatively simple to solve, like the unit disc or an annulus, then we have reduced the problem to solving the Laplace equation on $E$ and finding a conformal map $f:D\to E$.