The map shows the current state of the game board - one color for each player for his owned tiles. The tiles with * are "life tiles" - you receive them when the game starts and they grant you a greater bonus, so there's an incentive to destory the enemy's life tiles first.
As each participant in the network sees the entire blockchain, the players can see the "compiled code", essentially the EVM opcodes, but only after all the players already committed to the hash of their code and new players can't commit anymore. This is a "commit-reveal" method that allows the game creator to accept registrations for a specific bot without actually revealing the player's code beforehand.
The registration fee is indeed 1 ETH at the moment, but that's a placeholder. The winner gets almost all the pool of registration fees (winning a game with 5 players = 4.5 ETH).
claimTile is the API exposed to the bot contracts - it's a way for them to change the board state to signify they own the tile. Of course, as every operation, it costs some gas. So you can have a strategy of claiming as many as possible by using only claimTile, but you could also seek enemy tiles and claim them. It's a trade-off of gas.