I am finishing reading this book, and I strongly recommend it to any game designer.
https://images-na.ssl-images-amazon.com/images/I/51xIdTyakWL._SX313_BO1,...
It talks more about video games than board games, but it's the first book I read that explain the psychology behind why we play game. In summary, they say that games are there to solve 3 needs:
- Competence (complete challenges, defeat a stage, etc)
- Autonomy (Make decisions to reach certain goals)
- Socialisation (With other players or with NPC)
Enjoy!