Although it doesn't look like it, this board has the same set of pieces used in a regular game of chess. However, these pieces are arranged in an unknown order, unknown to both the players and to the computer. The identity of these pieces will be defined by the players, by playing the game.
This game is titled "Quantum Chess" because each piece is in a sort of superposition (a piece in a superposition will be called a superpiece). What this means is that every superpiece could be one of many options. And, when finishing the game, the moves that every individual superpiece made must be possible for one of its' options. In fact, every superpiece could have been represented by a single regular piece the whole time, without violating any rules.
The good news for you, the player, is that you don't need to keep track of what every piece secretly is. The game does that for you, by showing, for every piece, which ones it could be. At the start of the game, every superpiece is shown as a wildcard (★). If one of these superpieces were to move one step forward, it would be shown as ♚♛♜♟ because King, Queen, Rook, and Pawn are the only pieces that could have made that move. For the rest of the game, this superpiece is no longer able to make moves that would only be possible for a Knight or a Bishop.
This causes another mechanic that you need to pay attention to. If any superpiece is shown to be the Queen ♛ (as an example), and only the Queen, then none of your other pieces can be Queens. You're only supposed to have one! This causes many other interesting effects as well, where a single move can change many different pieces at once. It's up to you to figure those out!
If you've understood everything so far, you might be wondering how a check works, when it's possible to have multiple pieces that could be the King. The answer is fairly simple: every superpiece that could be a King must be in check in order for the player to be in check, and similarily for checkmate. It's important to consider that whenever a superpiece is captured, it retroactively cannot be the King, which can give you additional power over your opponent.