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)]