First show that $k^2$ snakes can be placed on a $(2k+1) \times (2k+1)$ chessboard. Divide the chessboard into strips of width $2$ (one strip of width $1$ remains). On any strip we can place $k$ snakes, one after another; so on $k$ strips, it is possible to place $k^2$ snakes.

It remains to prove that one can not place more than $k^2$ snakes on the chessboard. Write numbers $0, 1, 0, 1, \ldots, 0$ in the odd rows, and numbers $2, 3, 2, 3, \ldots, 2$ in the even rows. Notice that no matter how we place the snake on the board, it always covers numbers $0, 1, 2$ and $3$. Since all numbers $3$ are in the squares with even row and column numbers, there is exactly $k^2$ of them, hence there can be at most $k^2$ snakes.