Rachael has PTSD from being bullied. She likes certain anime because they offer some emotional solace and show great friendships.
The himedere comes from 'hime' or princess, + 'dere' as in 'love struck'. So, much like how the term 'tsundere' refers to someone who masks their love struck feelings with a rude, feisty attitude, the himedere masks them with the hauteur of a princess or noble.
Note that she may not be a literal princess, as many examples are Rich Bitch types. Also, many literal princesses are not this type. For example, the titular character in Sailor Moon is a princess, but she's more of the 'dere-dere' or 'genki girl' type.
Though at heart I'm a pearl, I'm a difficult girl, so buddy beware!
— Bonnie/Erma, 'Anything Goes'
Subtypes and Related Types
A male version of the himedere. From the word 'ouji' meaning 'prince'.
Not to be confused with a Ouija board.
Examples include Lelouch from Code Geass, Yami from Yu-Gi-Oh!, and Ciel from Black Butler.
This comes from 'kami' or 'god'. A character similar in arrogance to a himedere or oujidere, but they are a god, act like/think they are a god, or have god-like supernatural powers. Note that some characters might be literal gods or goddesses without this arrogance, such as Belldandy from Ah! My Goddess! This trope is more about an arrogant and forceful personality, regardless of actual power or god-status.
Examples include Light from Death Note, Ragyo from Kill La Kill, and Gendo from Evangelion.
Well-Known Anime Examples
Nagi from Hayate the Combat Butler, pictured above.
Ayeka from Tenchi Muyo!, the top picture of this article, may be the first example of this, and codifies some of the tendencies.
Vegeta in Dragon Ball Z is an oujidere, being a Saiyan prince who looks down on other "lowly" characters.
Boruto starts out with a haughty attitude, because his parents are famous.
Kimblee in Fullmetal Alchemist: Brotherhood.
Inuyasha's Sesshomaru is like this and a kuudere.
Asuka in Neon Genesis Evangelion is a combination of this, the "fiery redhead" type, and a tsundere (minus the 'dere'). As an elite Eva pilot, and a brilliant prodigy, she thinks highly of herself. Which is why she has a such a crappy attitude initially towards almost everyone else, especially Shinji.
Tons of high school romance comedy or drama anime have the himedere as a stock character. Usually she's rich but not royalty. But don't let that fool you.
You'd think Emilia would be like this in Re:Zero, but her history - as a victim of racism and unfounded suspicion - has made her humble. Beatrice is like this, with the princessly 'twin drills' hair. She's also sort of a tsundere, to Subaru. When she's not being the awesome Badass Bookworm that she usually is.
Nui Harime in Kill La Kill has shades of this, until she goes full yangire. Ragyo and Satsuki are like this and are also the 'kamidere' type, with oh, so little dere.
Examples Outside Anime
Rarity (pictured above) in My Little Pony: Friendship is Magic is a himedere, but interestingly enough, the actual princess characters are very humble.
Lots of high school drama movies and TV shows have a spoiled, rich girl or guy who could be called a himedere or oujidere, if they drop the royal attitude due to a crush or friendship.
Veruca Salt from Charlie and the Chocolate Factory. She's used to getting everything she wants, and it shows.
Draco Malfoy from Harry Potter. He sees muggle-born or half-muggle wizards and "lesser" families like the Weasleys as lower than his "prestigious" pure-blooded wizard family. And acts much like an anime oujidere.
Bonnie/Erma from Anything Goes, quoted above.
Georgette (and basically any fictional poodle in any show with talking dogs) in Oliver & Company.
Why is this type appealing?
Honestly, much like a poorly written tsundere, a poorly written himedere just makes me want to bash their face in with a rock.
But to people who aren't me, why is a stuck-up rich bitch appealing?
Maybe because she is hard to please and has high standards, like the tsundere. Then it feels more rewarding when we watch her fall for someone and lose the arrogance. It's similar to the kuudere, we like seeing them drop the act and admit their true feelings.
It's also satisfying seeing this type of character fall from grace or be forced to eat a bit of humble pie. And hey, without the character being imperfect to begin with, we can't have character growth.
Anime is not unique in using an upper class, snobby character as a stock villain or rival to the main character. But anime usually shows such a character's more vulnerable and tender side over time. Often, their haughty attitude is shown to be the result of strict, overbearing parents. For example, this is the case with Satsuki in Kill La Kill. They are also susceptible to starting out evil and turning good when they are beaten or other circumstances bring out their more sympathetic side, for example, the Digimon Emperor in Digimon.
Rare and interesting are works which center on the personal struggles of himederes. For example, the manga Ojojojo is a comedic, intimate look at this trope, showing the social struggles of a himedere-type protagonist. Works like this show that the himedere has issues with wanting to live up to their family name or prestige or political position. At the same time, they're just ordinary children or teens trying to navigate the same social minefield we all have to navigate. So they use hauteur as a mask to avoid showing their real vulnerability - and some of them are actually really sweet underneath!