Like many proofs by contradiction, I find that this proof that there is no knight's tour on a 4xn chessboard hides a bit of what's really going on. Here's a more direct version of the same proof that I find more perspicuous. Call the squares in the top and bottom rows of a 4xn board the "extreme" squares. Since you can't get in one knight's move from one extreme square to another, the maximum density of extreme squares in a knight's tour is 50%, and this is only attained if every other square in the tour is extreme. Since half the squares on the board are extreme, a tour that includes every square must alternate extreme with non-extreme squares. But a knight's tour alternates white and black squares, so a tour that alternates in this way cannot reach both black and white extreme squares. QED

Stop reading here, if you so desire. Otherwise: first let's prove the seemingly unrelated fact that, on a standard 8x8 chessboard, the most knights you can put on with no two attacking each other is 32. It's easy to see how to attain 32: cover all the white or all the black squares with knights; each knight move changes color, so these 32 are clearly non-attacking. Now to show that these are the unique two configurations that attain this maximum: we already know that there is a knight's tour on the 8x8 board (a picture of Euler's solution is included in the article, for example). Given any two squares which are used consecutively in the knight's tour, at most one of them can be covered by a knight from a "no two attacking" configuration, of course. Therefore the only way to pack 32 knights is to place one on every other square in the knight's tour -- which means all the black or all the white squares, since again, knight moves alternate colors. Nice proof of the wrong theorem. But now apply the same proof to the 4xn board, and we see that if a knight's tour existed, then the only way to fit 2n nonattacking knights on a 4xn board would be to use all the squares of a single color. But that's manifestly not the case: on a 4xn board, cover the entire top row and the entire bottom row with knights; 2n of 'em, no pair attacking. QED.