Roughly speaking, a function $f:\mathbf R^2\to\mathbf R^2$ is differentiable at $\mathbf v$ if there is a linear transformation $Df(\mathbf v)$ such that $f(\mathbf v+\mathbf h)\approx f(\mathbf v)+Df(\mathbf v)\mathbf h$. If, in addition, $f$ is differentiable as a map from $\mathbf C$ to $\mathbf C$, then we require that this linear transformation is complex multiplication, which rotates and scales the vector $\mathbf h$. This means that if you consider a tiny disk $D$ centred at $\mathbf v$, then (in the limit) the image of $D$ under $f$ will also be a disk.
My question is: from a geometric perspective, why does the fact that holomorphic functions map disks to disks give them such nice properties—e.g. a function which is differentiable once is infinitely differentiable.