A simple explanation is by thinking of the determinant expansion as a sum of products. Say a row is repeated, call its members A1, A2, .... An.
In the determinant expansion, we can match the n! products in pairs:
I) the term (terms from first rows) x Ai x (terms from the rows in between) x Aj x (terms from remaining rows)
II) the term (terms from first rows) x Aj x (terms from the rows in between) x Ai x (terms from remaining rows)
Observe that everything from I and II is matched, except "i" and "j" are switched. It takes 2*|i-j|-1 adjacent switches to interchange "i" and "j".
That's an odd number SO the corresponding permutations are of opposite parity (if one is odd the other is even and vice versa); so the sign in the expansion for I and II is opposite, but magnitude is the same.
THEN, those two terms add to ZERO. Ergo everything, having been split into pairs that add to zero, adds to ZERO. q.e.d.
(the difficulty for physics types, like me, is seeing the "source" of the property that the determinant switches sign when you swap columns/rows. why to swap two elements in a row you have to do an odd number of exchanges/neighbor swaps? You have to move BOTH of them over all the intervening elements - giving you an even number of neighbor swaps, and then ONE time over each other, making the total number of neighbor swaps odd.)