I've marked up this board a bit to show why this is checkmate, showing all the attacks that make it one. Lets go through them one by one:
- The pawn - The black king is currently in check by the white pawn on the diagonal. The king would need to move away, or the pawn be taken to remove this.
- The rook - The black king is in check because of the white rook at the bottom of the same column. The king would have to move out of that column, another piece would need to block, or the rook be taken to remove that check.
- The bishop - The black king can't move into the corner to get out of check because the white bishop is threatening that square.
- The knight - The black king can't take the pawn to get out of check by the rook and pawn because the knight is threatening that square.
There's no single move here that will get the king out of check. If the pawn is taken by another piece, the rook is still holding the king in check. If the king moves to the corner, out of check by the rook and pawn, he is now in check by the bishop and if the king takes the pawn, moving out of check from the rook at the same time he is in check by the knight.