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To $f_i$ corresponds an element, say $F_i$, of $S_{d_i}$. Multiplying by $F_i$ one gets complexes $0\to S_m \to S_{m+d_i}\to 0$. Now try a tensor product of such complexes.
One should think of the "space of G orbits" as a first approximation of "taking the quotient by the G action". Then it is really helpful to have a finitely generated ring of invariants.
Going to my Home Page to see that old paper of mine my impression is that Merkurjev uses Matsumoto's theorem. But notice that nowadays one also has the Milnor-Witt K-theory approach.
