We can solve this by using either rows or columns approach. Lets do by rows. For 1st row we, have 8 options to place the 1st rook. For 1st row, we have 8 choices, for 2nd row 7 choices and so on...so 8! = 40320.
Note that we should not count possibilities by BOTH rows and columns as they give us the same set of 8!