Firstly,
When you have 2 things multiply by each other and the product is zero, if the first (x) is not zero, the second (A) must be zero. Determinant, loosely speaking, is finding the magnitude of the matrix A.
As for why at least one all zero row vector, this is not always true. A matrix with at least one all zero row vector will gives determinant zero, but having determinant zero does not mean it has one all zero row vector (eg 2by2 matrix with all element 1).
Sorry my bad. Yes the product of 2 non zero matrix can be zero. see (Multiplication of Matrices) if need example. In the original question, we need the determinant of A to be zero for AX=0.
One of the way of calculating det, is to reduce it to reduced echelon form. Now we need det(A)=0 if AX=0 for nonzero x.
In order for det(A)=0, one of the element must be 0, as long as at least one of the element in the main diagonal is zero in reduced echelon form. In order for 1 element in the main diagonal to be zero after reducing, at least 1 row or 1 column must be zeroes in the original matrix.
Another way for calculating det is to use the cofactor expansion in which we will choose the most convenient row/column, which in this case, is the all-zero vector row/column, which will give us det(A)=0 as well.