This is essentially selecting $r$ people from $n$ men and $m$ women.
i.e
Select $r$ people from $m + n$ = $\binom{m+n}{r}$
which is equal to $\Rightarrow$ (Select $0$ men and Select $r$ women) OR (Select $1$ man and Select $r-1$ women) OR $\cdots$ (Select $r$ men and Select $0$ women).
$\Rightarrow$ $\binom{n}{0}\binom{m}{r}+\binom{n}{1}\binom{m}{r-1}+... +\binom{n}{r}\binom{m}{0}$
$\therefore$ $\binom{m+n}{r} = \binom{n}{0}\binom{m}{r}+\binom{n}{1}\binom{m}{r-1}+... +\binom{n}{r}\binom{m}{0}$