Let $X$ be a complex manifold with an action of $G=GL(n,\mathbb{C})$ which is free and locally proper (each point of $X$ has a $G$-invariant neighborhood on which $G$ acts properly.)

Satz 24 of the paper

H. Holmann, Quotienten komplexer Ra ̈ume, Math. Ann., 142 (1961), pp. 407–440

asserts that $X/G$ is a complex manifold.

Why is it true?

The biggest issue is, since I cannot read German, I don't know if by "free" he means set-theorecial free or scheme-theoretical free (the latter is in the sense of Mumford's GIT book.)