There are several solutions to this puzzle, but they are all quite similar.
Here's a possible one:
Proof of maximality 
There are 15 diagonals on the chessboard running from bottom left to top right. They are:
Each of these diagonals can only contain one bishop. Also, the first and last diagonals cannot both contain a bishop, since both are on the diagonal a8-h1. Therefore, we can place at most 13 bishops on the other 13 diagonals, and one bishop on those two diagonals, for a total of 14 bishops. Since 14 bishops is possible, 14 is the maximum number of bishops we can place so no two attack each other.