Bannai, Bannai and Stanton proved that $f_d(k) \leq {d + k \choose k}$ in 1983. See: http://link.springer.com/article/10.1007%2FBF02579288
I don't think this bound has been improved in general. It is certainly not tight for every value of the parameters.
The first known bound for this function was for $k = 2$, as given by Larman Rogers and Seidel in 1977, "On Two-Distance Sets in Euclidean Space". They proved that $f_d(2) \leq (d+1)(d+4)/2$ using a nice dimension argument. This bound was later improved by Blokhuis in 1981 to $(d+1)(d+2)/2 = {d+2 \choose 2}$, which was later generalised to $f_d(k) = {d+k \choose k}$.
This problem is also discussed in the manuscript "Linear Algebra Methods in Combinatorics" by Babai and Frankl, where you can find some of these proofs.