This is the whole idea in one picture. The black king on h8 is checked by a knight on f7, and he is surrounded by his own furniture: rook on g8, pawns on g7 and h7.
8 
7 
6 
2 
g Notice what is not happening. Nobody blocks, because there is never a square between a knight and the king it checks. Nobody captures, because the rook on g8 does not see f7 and the king cannot reach it. Black is up a queen on a5 and it is completely irrelevant.
That is a smothered mate: a knight, a boxed-in king, and no way out. The knight is the only piece in chess that can deliver it, for the same reason it is the only piece that jumps over other pieces.
Philidor’s legacy: forcing the box shut
In the diagram above the rook was already on g8. In a real game it is on e8 or f8, and the beautiful part is that you can make Black put it there himself. The sequence has a name, Philidor’s legacy, and it goes like this.
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3 2 g Walk it through. 1.Qe6+ hits the king along the diagonal and he must go to h8, because 1…Kf8 runs into 2.Qf7 mate with the knight on g5 guarding the queen.
2.Nf7+ Kg8 is a check the king can only answer by stepping back. Then 3.Nh6+ is the move that does the work: it is a double check, from the knight on h6 and from the queen on e6 down the diagonal that the knight has just vacated. A double check cannot be blocked and cannot be met by a capture, so 3…Kh8 is forced.
4.Qg8+ is the sacrifice. The queen puts herself where the rook must take her, and 4…Rxg8 is Black’s only legal move. Now the rook is standing on g8, the king’s last square, and 5.Nf7 is mate.
Where you will actually meet it
Look for it whenever an enemy king has castled short, its rook is still on f8 or e8, and you have a knight that can reach e5 or g5 with a queen on the a2 to g8 diagonal. Those three ingredients are the trigger. In my own games it has come up perhaps a dozen times in twenty years, always in the same shape.
Even when the full combination is not there, half of it often is, and a knight check that forces a king to shuffle can win a tempo you need for something else. Run a set of smothered mate puzzles until the shape is automatic, then compare it with the back-rank mate, which is the same crime committed by pawns instead of pieces.