This is a theorem made in Kodaira's Complex Manifolds and Deformation of Complex Structures Chpt 1, Sec 1.2, Thm 1.20. I understood the proof is done by reparametrization and showing independence of partial derivatives but I still have a hard time to convince myself that the statement is intuitively clear.
Thm 1.20 Let $f_1(z),\dots, f_m(z)$ be holomorphic in a domain of $C^n$. Suppose $rank(\frac{\partial(f_1,\dots, f_m)}{\partial(z_1,\dots, z_n)})=v$ independent of $z$ in domain. If $z_0$ is a point of this domain s.t. $det(\frac{\partial(f_1,\dots, f_v)}{\partial(z_1,\dots, z_v)})\neq 0$ at $z=z_0$, then there exists neighborhood $U(z_0)$ s.t. $f_{v+1},\dots, f_{m}$ are functions of $f_1,\dots, f_v$.
$\textbf{Q:}$ What is relation between submanifold via constant rank($f_1,\dots, f_v$ defines a submanifold) and the consequence that $f_{v+1},\dots, f_m$ are functions of $f_1,\dots, f_v$ if one can deduce conclusion from submanifold here? In other words, what is geometric interpretation of the conclusion? It seems that the statement itself has nothing to do with holomorphicity. Hence, I believe the statement is true for real smooth functions as well.
Note that in a neighborhood of $z_0$ we have $df_j\wedge df_1\wedge\dots\wedge df_\nu = 0$ for any $j=\nu+1,\dots,m$ (or else the rank would be at least $\nu+1$). This means that $df_j$ is a (functional) linear combination of $df_1,\dots,df_\nu$, which in turn means that $f_j$ is a function (on that neighborhood) of $f_1,\dots,f_\nu$.