Why do we need a restriction of a game to prove the given statement?
It is because the set of strictly dominated strategies changes as we eliminate strategies.

The point of the proof is to show the result for ANY $R$ such that $R \rightarrow_{S}R'$.

Look at Example 7 in the next page of the paper. 

In $S$, there is only one strictly dominated strategy, and we remove it and get $R$ (hence $S \rightarrow_S R$).
Now look at $R$. There is a strictly dominated strategy that was not one before (in $S$), removing it get $R'$ (hence $R \rightarrow_S R')$.