In short, wearing gauze, casts, eyepatches and similar items for Fetish Fuel purposes, or seeing it as such.
Particularly well-entrenched as a Moe trait in anime/manga circles, and featured in the cosplay style/fashion trend called kegadoru ("injured idol"). The basic psychology isn't difficult: the appearance of injury implies the need to be protected, which inspires a desire to approach and comfort the injured person. The fact that wearing only bandages implies exposure, deliberate or accidental, doesn't hurt either.
A more extreme version of this fetish is abasiophilia, a sexual obsession with disability or the appearance of disability. Amputations, prosthetics, parapalegia, and the like fall under this category.
Sarashi is a sister trope. An older sister trope, actually. See also The Florence Nightingale Effect, which is when a nurse or doctor falls in love with a Bandage Babe. Usually involves a Bandaged Face. Overlaps with Good Scars, Evil Scars.
Actually a case of Beam Me Up, Scotty!, aside from some of the box art, Lucy as a child in a hallucination note the one where the bandages fall down to show Lucy the voice talking to her has her own face, and Mariko in her first appearance, the diclonii don't wear bandages, or usually anything else for that matter.
Takatsu Aoba from Code:Breaker has her right arm all bandaged up and wears it in a sling after Ogami sliced it off.
Kagerou/Mishiru from the same series,is bandaged to the point hers gender is unnoticed if you're familiar to her.
Especially noticeable when Abiru is used by Chiri as an example of perfectly half-naked. Yeah.
She's really a parody of this trope, since the bandages cover injuries she brought upon herself by repeatedly bothering rather dangerous animals due to her tail fetish. Also she even admits she just puts them on even when she isn't injured as she likes the attention.
Although in the final OVA, Bangaichi, she does get labeled as "Ayanami" in a scrolling screen gag.
Subverted by Adorea of Franken Fran. A female servant of Fran's covered in bandages like a mummy. Her striking figure, beautiful hair, and selfless willingness to give her organs certainly made her look like a bandage babe, and the volume cover that has her story even made it look like that underneath those bandages was a naked, classic beautiful girl. Then, when a terrorist who was smitten with her asked to see her face, she removes the bandages around her head and we're treated to a frame filled with nothing but tentacles surrounding a gaping maw that she uses to swallow the terrorist whole.
In Fullmetal Alchemist Lan Fan is injured and cuts off her own arm after an attack from King Bradley and for the majority of her appearances before she leaves to get a replacement arm she's pretty much topless except of the bandages around her chest.
Game X Rush gives us two male examples in the last chapter, as well as a single-panel flashback of Miyuki as a Bandage Babe.
Rather common on Bleach since everyone spends so much time having been nearly slashed, stabbed and chopped to pieces. Ichigo takes it so far that even his sword is wrapped in bandages!
Full Metal Panic! has a fanservicey male version of this with Sousuke, in the later parts of the story. Nicely illustrated in the Sigma manga. After being in a coma for a month, Sousuke was rendered very weak and helpless in bed, being half naked and having his bare chest and arms wrapped in bandages. (Of course, "helpless" for him means only being able to bomb the hell out of the enemies that try to ambush him.)
Shizuka/Serenity from Yu-Gi-Oh! is going blind for a period early in the series, and wears bandages that completely cover her eyes after her eye surgery.
Ruthlessly subverted with Balalaika in Black Lagoon. One of her flashbacks involves her sitting in a vehicle with bandages covering her burns and wounds from a recent battle. There was nothing remotely sexy about that scene as she looked exactly what you'd expect from someone coming from a war zone. She also retains a large number of scars over her body because of those wounds.
Mikaila from Godchild is completely covered with bandages (except for her head) under her clothes, due to the fact that she is literally rotting away underneath due to a bad case of clone degeneration.
Rokujo Chikage from Durarara!!. Evidently, having his face violently rearranged by Shizuo has done nothing to hinder his chick magnet status.
Gunslinger Girl. A flashback scene shows Henrietta like this before she became a cyborg, when she was in hospital after a brutal attack. Becomes squick when a full-body view shows that one of her legs has been amputated. Triela is also shown this way after her fight with Pinnochio (it's more of a fanservice version as Triela has only got her skirt and bandages on under the blanket, and she gets a Cool Down Hug from her male handler). Due to the nature of the cyborgs she gets better after a quick operation.
Yuuta also did that during his middle-school delusions.
Seems to be an instance of Author Appeal in the manga Karneval, as characters commonly appear on the volume covers in bandages. Karoku is a notable example in that he wears bandages around his head and torso for most of his in-story appearances, for reasons that are currently unknown.
A rare male example in Loveless with Soubi, who wears bandages around his neck because the carvings on it bleed when he fights.
Another male example is Zange Natsume of Inu X Boku SS, who wears bandages over his right eye. Not because he can't see out of it though, as we later learn.
Rare Male Example: Kuranosuke Shiraishi from The Prince of Tennis, who wears bandages all over his left arm. There's a Running Gag of him either starting to take them off or threatening to do so, but it never happens. The manga reveals that the bandages actually hide a heavy bracelet/arm guard that acts as his Power Limiter.
After almost dying in the battle against Merak in the Devil Survivor 2 anime the already Stripperiffic Hinako spends the rest of the series with a bandaged eye and arm.
Snow White And Seven Dwarfsloves this trope, to the point where it'd be easier to name the characters who don't wear or end up in bandages at some point.
YuYu Hakusho: Mukuro double subverts this as her bandages conceal a face covered in horrific burns and scars, over but she still has plenty of ship tease with Hiei in the anime and a clear bond with him in both adaptations.
Cleo from Monster High wears bandages, being a Bandage Mummy, as does her sister, Nefera. Cleo at least, and possibly both of them, is cursed to have to wear some of her bandages at all times or suffer some unexplained consequence, but she doesn't have to wear all of them, leaving all her outfits to end up like this. Rule of Sexy is enforced almost constantly.
Warhammer 40,000 has the Sisters Repentia, members of the Adepta Sororitas who have forgone their armour and firearms in order to undergo penance for their sins, and charge into battle wearing nothing but ragged strips of cloth and wielding chainsaw-swords almost as tall as themselves.
Female (and male) stars in The Movies recovering from chest augmentations wear these on top, and pajama pants on the bottom. The same set is available as a costume option for use in movies.
In Soul Calibur 4, female custom characters have the option to choose a layer of bandages as their underwear.
Xelors in Dofus are covered in bandages. Probably for good reason, as Wakfu reveals that they aren't even strictly organic...
In Neverwinter Nights, when you first meet Aribeth, she is scar free, and her army is shiny and at %100. However, When she turns to the bad guys, she gets new armor, a new model, and a new portrait. You can see a heavy bloodstained bandage around her waist through the part of the armor that normally would have shown some skin
Viola Cadaverini from the Phoenix Wright: Ace Attorney series. She considers herself the "lover" of Furio Tigre, although it's debatable as to exactly what they've been up to. Given that he's flat-out terrified of her, the answer's probably 'whatever she wants'. That bandage is the reason Furio Tigre had a debt himself.
Defendant Lana Skye cut her hand, supposedly while she was attacking the victim with a knife. You find out about this when the otherwise stern Lana holds up her bandaged hand to cover her mouth when she gasps in surprise at something said.
Kay Faraday also ends up covered in bandages after she gets attacked in Ace Attorney Investigations 2
Kainé in NieR keeps her left arm and leg covered by bandages constantly. It's never revealed what's under them, but it's implied they were scarred in a Shade attack. Also, her bandages cover more skin than her actual clothes.
Bandage Girl in Super Meat Boy can be seen as this taken to ridiculous extremes.
Hanako from Katawa Shoujo may count, due to an artist pin-up found for her evolution over the years.
Super Dangan Ronpa 2: Mikan Tsumiki is a strange example of a nurse who also does double duty as a Bandage Babe. She's massively clumsy, explaining it somewhat. Gundam has a bandaged left arm, but that's probably due to his status as an animal breeder... Or his occasional Chuunibyou traits.
Anghel Higure's human design in Hatoful Boyfriend. The one on his chest makes sense— he's actually a bleeding-heart dove—but the eyepatch and arm bandaging is probably just for this trope.
He does talk about having a "Demonic Red Eye". ...Anghel is confusing.
Caren from Fate/hollow ataraxia, not visible in her usual clothing (the most you see is some wrapped around her hands), however underneath most of her body is covered with bandages.
Caren's mother Claudia was this, too. She wore an eyepatch and bandages, due to being a mixture of Ill Girl and Blessed with Suck.
Hank and the two new protagonists in Madness Combat both wear bandages, (Hank to the point where he's almost a Bandage Mummy) These seem to be used both to make them look badass and to distinguish them from other mooks