This is "due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane", to quote T.W. Korner on the subject.

I am informed that the proof I met is prepared using:

J.-P. Kahane *Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et les sommes algebriques d'arcs croissants* in the conference proceedings *Harmonic analysis, Iraklion 1978* Springer 1980

G. G. Lorenz, *Approximation of functions* Chelsea Publishing Co. 1986 (First Ed. 1966)

A. G. Vituskin *On the representation of functions by superpositions and related topics* in L'Enseignement Mathématique, 1977, Vol 23, pages 255-320

[This is all from these skeleton notes (no proofs) here (Links to pdf; See Chapter 1 and Chapter 11 for references)]