First note that:
If two pawns are on the same column, then placing a pawn in the same row as either of them forms a right triangle. I'll call any pawn in such a pair grouped.
Because there are 8 columns and 15 pawns:
The maximum number of pawns that are not grouped is 7. 8 is impossible, since there are only 8 columns and this would mean you would place at most one pawn in each, leaving nowhere to place the other 7.
This means that:
There are at least 8 grouped pawns. If there were two of those in the same row, they would form a right triangle, so they must be all in different rows. But then you can't place any of the remaining 7 pawns without forming a right triangle somewhere, since all 8 rows must have at least one grouped pawn.
This bound is actually tight, since we can place 14 pawns on the chessboard while not forming any right triangles: