As per joriki's suggestion, I may as well post my above comment as an answer.
The idea is to think about how to detect even a single $1$ entry. If the job was to detect the presence of a $0$ in a row, then $x_{i1} x_{i2} x_{i3} \cdots x_{in} $ would do the job - it would be $0$ if and only if any entries were zero. So then after thinking about it for a few seconds (or as I did, a few minutes) we realize we can detect $1$'s in the same way, by instead considering
$ \prod_{j=1}^n (1-x_{ij}) =(1-x_{i1})(1-x_{i2})...(1-x_{in}),$ which is $0$ if and only if any of the entries in a row is $1$, and is $1$ otherwise (that is, if all entries are $0$).
So then the number of zero rows is given by $ \sum_{i=1}^m \prod_{j=1}^n (1-x_{ij})$, meaning the number of non-zero rows is equal to $m -\sum_{i=1}^m \prod_{j=1}^n (1-x_{ij}).$