You simply can imagine it as a checkered square field and after you cut out a row or column you put the pieces back together and get the same, but just smaller square field. It perfectly shows why the order doesn't matter.

Look at it this way. A configuration is valid if no two rooks share the same column or row. So given that there are n rows and n columns, and a rook uses up one row and one column, it doesn't really matter which row and which columns it uses, since you will have left n-1 rows and n-1 columns. Notice that once you are actually solving the problem, the position actully matters while counting because you might count choosing [(3,1) then (2,4)] and [(2,4) then (1,3)] as diferent configurations, but that's the fun part of the problem.