It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this note. When $d=1$, the statement is an easy exercise: one can write $f_1= da, f_2=db$ with $(a,b) = R$ since $R$ is an UFD.
There are a few proofs of this very interesting theorem, analytic (the original one, over complex numbers), using duality theory (Lipman-Sathaye, for all regular rings) and reduction to characteristic $p$ + tight closure (Hochster-Huneke, when $R$ contains a field, see the note in the first paragraph for some exposition of this method) but I am not sure any of them can be called direct. Any new proof will be very exciting!
