The move involves Zelda summoning a Phantom, a disembodied guardian suit of armor from The Legend of Zelda series, which attacks the opponent and disappears. It can be charged up to increase the range of the move. The Phantom is a solid object in the environment, so it can block movement and attacks of other fighters.
Phantoms first originated in The Legend of Zelda: Phantom Hourglass, as guardians of a temple that Link must visit at various points throughout the game. They are shown to be indestructible for a large portion of Phantom Hourglass, and getting attacked by one would result in Link being sent to the beginning of the temple. Phantoms in this game come in three variations, each with their own abilities.
They reappear in The Legend of Zelda: Spirit Tracks with the same function, although this time they guard the Tower of Spirits and now come in four variations. After Link and Zelda head to the Tower of Spirits, they are ambushed by Chancellor Cole and his apprentice Byrne who take away her body with plans to resurrect the evil Malladus. At some point, Zelda's soul becomes trapped in the Phantom, which results in her controlling its body, and she and Link must make use of the unique abilities of each Phantom.
This mechanic proved vital to proceed through Link's quest in restoring the tower and freeing Zelda's original body from Malladus' possession. As Zelda has control of the Phantom in SSB4, the move is most likely referencing Spirit Tracks.
The Phantom draws inspiration from the Zelda games for the Nintendo DS, though with some notable differences:
The Phantom has three different attacks, based on the charge of the move: uncharged, the Phantom does a quick and weak inwards slash; mid-charged, it does a stronger outwards slash; at maximum-charge, it does a powerful upwards slash that hits twice. While the Phantom is being shot out by Zelda, it can be reflected, which sends it in the opposite direction and changes its ownership.
The Phantom is a solid object, so fighters cannot pass through it instantly, and it can be attacked, allowing it to act as a wall. If the Phantom takes 13% damage or more during one summon, it will break, at which point it will disappear and Zelda will be unable to use Phantom Slash for another 9 seconds; a purple cloud aura will appear over Zelda's hand for the duration of this period.
Due to it acting as a solid object, making use of this move allows Zelda to control space on the stage, and defend against opponents' attacks or projectiles. Opponents passing through the Phantom (including rolling or air dodging through) can be punished by Zelda due to the relatively low end-lag of the move. The uncharged version of the move is particularly fast, allowing Zelda to summon the Phantom to force the opponent to react, while still having time to potentially shield or dodge counter-attacks.
Special Move customization was added in Super Smash Bros. 4. These are the variations:
The Phantom works slightly differently from Smash 4. In Ultimate, when charging, the Phantom appears behind Zelda and assembles itself. The move has six different different charge stages. In order: a kick, punch, an outward slash, a downward slash, and an upwards slash. Once the Phantom is fully assembled at its sixth stage, Zelda is able to move and attack freely, while the Phantom momentarily remains still until attacking with a dashing sword slash. The Phantom can be destroyed by interrupting Zelda mid-charge or by dealing enough damage to the Phantom. If destroyed, the Phantom will remain broken on the ground for two seconds, but can be resummoned as soon as it disappears.
As of patch 3.1.0, it is actually possible (though technically difficult) to summon the Phantom in front of Zelda by making use of a cancelled RAR down aerial into the Phantom Slash. This advanced technique, which some call the "Reverse Displaced Phantom", allows the Phantom to provide the same kind of frontal protection it previously offered in Smash 4.