Well, first thing I do if I don't see why a move was considered wrong is put the position into a chess engine to check my analysis (unless it was a question for a positional plan). In roughly 2/3 of the cases it turns out I missed something. (Of course one could expect to get that explanation from the author of the lesson.)
Here Stockfish DD rates Kd2 about half a pawn better than Kd1 - not that much difference, but there isn't that much advantage to begin with. Of course given the topic and rating of the lesson I think any move that evades Black's mate threat and salvages an advantage should at least be an alternative move!
But there are worse mistakes in Chess Mentor lessons:
In "Silman Teaches Tactics (1)", lesson "Mirano-Castelliano", Silman expects you to prefer a mate in four instead of two different moves giving mate in three (7.f3+ instead of either Be2+ or Qg3+). Both shorter mates aren't even alternative moves but considered plain wrong.
And in lesson "Hug-Fegebank" one of the possible mates in two is considered wrong as well (3.Bc4+ gets a: "Not a bad check but after 3...Bd5 White will still have to find the crusher." Hmm, let's see: 4.Bxd5 and no matter what Black does 5.Qg7#...)
http://www.chess.com/chessmentor/view_lesson?id=6886
I played d1 and it told me it was an incorrect move. Then I played d2, which it accepted as the correct move.
Your result:
Score: 60%
Rating 1878 (-8)
I don't understand. But why is d2 better?
What if black plays Qa5? Wouldn't that pin the knight to the king? Surely trading queens wouldn't be a desired outcome. Our queen is well placed waiting to mate. We could also move our queen and threaten a check on the back rank but their knight has the back rank covered. Wouldn't d1 be the better move here so our king is out of the queen's way so we can still move our knight and check?