Yup I used bit sets to write a super duper fast Sudoku solver. For examples you can detect naked singles extremely fast by just and-ing the bitsets in a given row, column or house, taking the complement, and checking if a unique bit is set.
But they used to be essential for automating chess. A 64-bit word represents a board, and a set of words represents the positions of each type of piece, one word for white pawns, etc. Another word shows places threatened by the white pawns, and ORing them shows all the threatened places.