As noted in math.stackexchange, the theorem is in part 3 of this paper: Hilbert, D. (1893). Über die vollen Invariantensysteme. Math. Ann. 42, pp. 313–373.
From a quick overview, I think the idea is to first prove the theorem when the set of common zeros is finite, using elimination theory. Then he generalizes to infinite zero-sets. Everything is done with homogeneous polynomials.
Now to find time to read it carefully.