Let $X=\{x_1,x_2,\ldots ,x_n\}$ and $Y=\{y_1,y_2,\ldots ,y_n\}$. We can count all possible matchings inductively. There are $n$ ways to match $x_1$ with vertices in $Y$. Given this match, there are still $n-1$ ways to match $x_2$ with vertices in $Y$. Given the previous matches, there are still $n-2$ ways to match $x_3$ with vertices in $Y$, and so on. Ultimately, given previous matches, there will be $n-(n-1)=1$ way to match $x_n$. Thus, there are $n!$ possible matches.