I'm unsure of how to install dependencies (including what dependencies there are) and how to run this app.

Please provide a simple readme. Follow this template. Feel free to ignore: Running the tests, Deployment, Contributing, Versioning, Acknowledgments.

In at least one place (Board.tsx, line 65) we're mutating props: (this.props.grid[pKing.row][pKing.col].checked = null;) which violates React prop immutability.

In the chessApp reducer on lines 10, 11, and 12, we make a shallow copy of the state and mutate it.

var newState = Object.assign({}, state);
newState.grid[action.end.row][action.end.col] = newState.grid[action.start.row][action.start.col];
newState.grid[action.start.row][action.start.col] = null;