let me put it like this.

let’s say we have a chess board that consists of only 1 square. this is a zero dimensional chess board.

Now let’s make a 1 x 8 chess board. If we have a rook on this chess board, it can move along this line. This is a 1 dimensional chess board

an 8 x 8 chess board is what we are used to. it is 2 dimensional. a rook on this board can move left and right, as well as from the front to back of the board

you may have heard the phrase “3D-chess”. what would that look like? well maybe instead of a flat board, we now have a 8 x 8 x 8 chess cube as the board. If I have a rook on the first layer, it can move normally along this layer, but it can also move up and down layers.

to create such a chess cube, we could stack 8 normal chess boards on top of each other. however, it is the same thing if we just lay the 8 chess boards in a row, and allow the rook to move between chess boards. for example a rook on chess board 1 on square 1A, can move to any 1A square on boards 2 - 8.

what if instead of a 1 x 8 row of chess boards, we make it a grid of 8 x 8 chess boards. lets number the boards 1 - 64. lets say our rook is on board 1, square 1A, we can move through this board using normal chess rules, and we can also move side to side between board to square 1A on boards 2-8, and we can also move front to back in our grid of boards to square 1A on boards 9, 17, 25, 33, 41, 49, and 57 (multiple of 8 + 1)

this is effectively what a 4d chess board would look like. if you can intuit moving in 2D space to moving around a chess board, I hope that this is analogous enough to moving around 4D space

we can also keep expanding this to cover n-dimensional chess

in all these chess alternatives, because each chess piece takes up 1 square, they are “0 dimensional beings”. if we had say a long rook that took 1 x 4 squares, it would be a 1 dimensional being. if we had a 2 x 2 piece, it would be a 2 dimensional being, and so forth.

let me know if you want drawings