For anyone having troubles with this concept, read this as the golden rule.
We reduce problems to equivalently hard, or harder problems; so that the solution of the problem in hand, can be used to solve the harder problem.
Given: S is NPC.
Q → S, means Q is easier than S. Or, S is harder than Q. There's no rockbottom here. Q could be easy as heck, or almost as hard. Q could be NP, P, CFL, RL — anything. Can't conclude.
S → R, means R is harder than S. Here, we have a rockbottom. R is NP, because S is NP. R could be more than NP, but we know for a fact that it is at least NP.
Now, is it strictly NP, or undecidable, or not partially decidable, we don't know.
Hence, R is NPH (Because we know NP, but don't know if strictly in NP)