What does the black dot next to the chess puzzle mean?


Yes.
Black to move would mean solution is Bf3#.
White to move then Rg5 to prevent mate?
If it's White's turn to move, White is up a piece so lots of different moves work. For example Bg5 or h4. Puzzles wont have equally good first move solutions, so for that reason too it's a black to move puzzle.
Also books should have a solution section, so you can tell that way too.