Put a knight on e4 on an empty board and count its squares: d2, f2, c3, g3, c5, g5, d6, f6. Eight of them, every one two along and one across, and not one of them touching the square it left. The Laws never use the word L. Article 3.6 says the knight moves to one of the squares nearest to the one it stands on that is not on the same rank, file or diagonal, which is the same eight squares described backwards. The knight goes exactly where the queen cannot.
h 1 The horse nobody rewrote
Chess reached Europe from India by way of Persia, and on the trip most of the army was rebuilt. The old counsellor who shuffled one square diagonally became the queen, the elephant became the bishop, and both got their long modern moves in the 1470s, in Spain and Italy. The chariot and the horse arrived intact. The chariot slid along ranks and files and is now the rook; the horse jumped two and one, and still does.
So the L is not a clever design decision by anyone we can name. It is simply the oldest move on the board, and it happens to be the one move that no other piece copies.
What the L costs
Three things, and every one of them shows up in real games.
The knight changes colour on every move. From e4, a light square, it lands on a dark one: d2, f6, c3, whichever you pick. Which means a knight sitting on a light square cannot touch anything else on a light square this move, and your queen on d5 can sometimes ignore a knight two squares away.
It is slow. To go from e4 to e5, one square, takes three moves: Nd2, Nf3, Ne5. To cross the board from a1 to h8 takes six. A bishop does that in one.
And it cannot waste a move. Because the colour flips every time, a knight can never return to a square in an odd number of moves, so it can never hand the turn back to the opponent. Bishops and rooks can shuffle and lose a tempo on purpose. Knights cannot, and in a zugzwang position that is the whole game.
One more consequence of the L worth knowing: from a corner square a knight has two moves, from the centre it has eight. Hence the old coaching line about a knight on the rim being dim.
What the L buys
Jumping, for a start. The knight is the only piece that ignores what is in its way, which is why it can move on the very first turn of the game while the queen and the bishops are stuck behind pawns.
And forks. A knight attacking a queen is never attacked by that queen, because the eight squares it comes from are precisely the ones off her lines. A queen on d4 cannot touch a knight on f5 or c6 or e2. She has to move, and whatever the knight is also hitting drops off. That geometry is why the fork wins more material at club level than any other tactic, and why I hand beginners forks before anything else. There is a whole tour of the piece in how the knight moves, and a pile of examples in chess tactics for beginners.