A theorem due to Gelfand and Kolmogorov

I found a proof in these lecture notes by Garrido and Jaramillo. See Theorem 18. They also have historical references.

Dugundji's Topology has a very short, readable proof.

you can find the proof on page 289. its very readable.

Gillman-Jerison, Rings of continuous functions, Theorem 7.3.