Here is Tychonoff's original proof, for powers of the unit interval. He builds a complete accumulation point of a given infinite set by transfinite recursion along the index set. On page 772 of this paper one finds the formulation of the general theorem (in my translation): "The product of compact spaces is again compact. This one One proves this theorem word for word as in he case of the compactness of the product of intervals". Some authors (Folland, see comment below and Walter Rudin in his `Functional Analysis') credit \v{C}ech Čech with proving the general result but \v{C}ech's Čech's proof is the same as Tychonoff's and, based on a reading of his papers, I think Tychonoff deserves full credit for the theorem and its proof.
@Henno: not Fundamenta but Mathematische Annalen.
