Consider a residue class different from zero 0 and let a denote a representative of this class. Then a is not a multiple of d.
Since d is irreducible, g = gcd(a,d) is 1 or d. The second possibility is excluded since d does not divide a. So gcd(a,d) = 1, and by the theorem above, the class of a is invertible.
We conclude that all nonzero elements in R[X]/(d) are invertible. So S is a field.