Can someone explain why a row replacement operation does not change the determinant of a...

One way to think about it, using the property: $det(AB)=det(A)det(B)$:
Adding a multiple of one row to another is equivalent to left multiplication by an elementary matrix.
Let B be some $n\times n$ matrix, A be an n×n elementary matrix which acts as an operator which adds k copies of row i to row j. So applying that same row operation to B will result in the matrix AB. Then without a loss of generality, A has the form:
$\left[\begin{array}{cccccc}1& & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & \ddots & & & \\ & & k& & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right]$
where $a_{ji}=k$
The determinant of a triangular matrix is the product of the diagonal. A has a unit diagonal, so det(A)=1.
Therefore,
$det(AB)=det(A)det(B)=1det(B)=det(B).$