Look for something that never stops changing

A knight has up to eight moves from any square, and a closed route can wander anywhere on the board. Chasing the route itself will not get you far.

Look instead for a quantity that behaves the same way on every move, whichever of the eight you pick. If some quantity flips on every single move, then restoring it to its starting value takes an even number of moves.

Write the knight's square as coordinates $(r, c)$, each from $1$ to $8$. One move changes one coordinate by $2$ and the other by $1$.

Ask yourself what that does to $r + c$. Every move shifts it by $\pm 2 \pm 1$, an odd amount, so $r + c$ changes parity on every move without exception.