Who wears the cloak, and displacer beasts, "seem to be somewhere else."
Observers don't see two bodies. They see the body but are mistaken as to its location.
Displacement is described as a visual illusion in this edition. You would expect it to be countered by truesight, or by a spell or item that grants true seeing.
While you wear this cloak, it projects an illusion that makes you appear to be standing in a place near your actual location, causing any creature to have disadvantage on attack rolls against you. If you take damage, the property ceases to function until the start of your next turn. (DMG p. 158)
Truesight
A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic.
Displacement (MM p. 81 (Displacer Beast))
The displacer beast projects a magical illusion that makes it appear to be standing near its actual location, causing attack rolls to have disadvantage.
Displacer Beast (Description)
... takes its name from its ability to displace light so that it appears to be several feet away from its actual location
This is mechanically the same as the cloak.
Does True Seeing counter this? That's an interesting question. I'd rule yes.
... gives the creature... the ability to see things as they actually
are. For the duration, the creature has truesight (see above), notices secret doors, hidden by magic, and can see into the Ethereal Plane ...
The key to that ruling is that what displacement does is create a visual illusion.
Let's use some lore to fill in the blanks.
To understand what's going on with displacement versus invisibility some lore is helpful. This item, once called a Displacer Cloak, is from the original game. It predates the displacer beast's introduction.
OD&D (TSR, Monsters and Treasure, 1974, Gygax & Arneson)
Displacer Cloak: This device warps light waves to make the wearer appear to be up to 1" away from where he is actually standing. It makes it 10% harder to hit its wearer, whether by smiting or Wand (add +2 to defense and saving throws). {1" = 10 feet in that editions scale convention}
OD&D (TSR, Greyhawk, 1975, Gygax/Kuntz)
DISPLACER BEASTS: {snip} They always appear to be 3' from their actual position, so for purposes of hitting them, or when they must make saving throws, they are treated as a man wearing a displacer cloak as far as penalties and bonuses are concerned.
AD&D 1E (Monster Manual)
The molecular vibrations of the displacer beast are such that it always appears to be 3' (left, right, ahead, or behind) its actual position. Thus, these monsters always cause opponents to subtract 2 from attack dice rolls and add 2 to their dice rolls for saving throws.
AD&D 1E (DMG)
Cloak of Displacement: This item appears to be a normal cloak, but when it is worn by a character its magical properties distort and warp light waves. This displacement of light waves causes the wearer to appear to be from 1' to 2' from his or her actual position. Any attack by missile or melee strike which is aimed at the wearer will automatically miss the first time*. Thereafter the cloak affords +2 protection, i.e. 2 classes better on armor class, as well as +2 on saving throw dice versus attack forms directed at the wearer (such as spells, gaze weapon attacks, spitting and breath attacks, etc. which are aimed at the wearer of the cloak of displacement). This can apply to first attacks from multiple opponents only if the second and successive attackers were unable to observe the initial displacement miss.
AD&D 2e had a similar description, though for displacer beasts there was a die roll to see if the beast was in front of, to the right or left of, or behind its perceived location.
A spell or item with true seeing made it possible to see
displaced creatures, like displacer beasts or folks wearing displacer cloaks. So would "Dust of Appearance."
Displacement was a psionic power in AD&D 2e, that worked like a
displacer beast's natural ability in making the character appear up
to 3 feet from her actual location. (Some Githyanki had this
ability).
D&D 3.5. (D20 SRD)
the cloak works like a displacement spell, and "its magical properties distort and warp light waves"
Displacement (A spell, 3rd level, Sorcerer / Wizard, D20 SRD)
The subject of this spell appears to be about 2 feet away from its true location. {note; displacement does not prevent being targeted, as total concealment would}
In this edition also "true seeing" reveals its true location
Summary
In all editions, displacement means that they see you, but they think that you are somewhere else due to an illusion, a magical effect, or a psionic effect that messes with light / vision.
