I had the same question and here is what I think the writer meant:
Consider choosing just doing inclusion exclusion on $I$. You get a factor of $(-1)^{I+1}$. Then you have to do inclusion exclusion on $J$, there you get a factor of $(-1)^{J+1}$. You multiply then to get $(-1)^{I+J}$.
I had the same question and here is what I think the writer meant:
Consider choosing just doing inclusion exclusion on $I$. You get a factor of $(-1)^{I+1}$. Then you have to do inclusion exclusion on $J$, there you get a factor of $(-1)^{J+1}$. You multiply then to get $(-1)^{I+J}$.