In the LHS $(}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )$ counts the number of ways of selecting $r$ objects from a set of size $n$ where order is not significant and repetitions are allowed. So you have $n$ people you form $r$ teams and select $r$ captains and select $(n-r)$ players.

The RHS divides up a team into $$2$$ sets?