It can be played on a regular flat chess board. But you have to remember that the right edge is connected to the left edge.
Pieces can go off the board to the right and end up on the left, and vice versa.
In the diagram on the left, we see that the Bishop on f1 is pinning the Knight on c6, and it is also attacking the Pawn on h7. Also the Knight on h4 is attacking squares b3, b5, and a6, as well as the squares it would normally attack from h4. An easy way to play cylindrical chess is to play with two boards side by side, and make the same moves on both boards. Then you can see that a Bishop or Knight on one board attacks squares and pieces on the other board. When I was young, I thought up a problem in cylindrical chess.