Kara Kara Bazaar has existed for at least one hundred years as a picture of it taken Princess Zelda before the Great Calamity is permanently recorded in the Sheikah Slate's restored photo album. Link and Zelda once visited Kara Kara Bazaar during that time as Link can recover one of his memories here as part of the "Captured Memories" quest. Presumably like Gerudo Town, its remote location in the desert allowed it to survive the Great Calamity.
It is the place where Link encounters Vilia during the main quest "Forbidden City Entry". As Gerudo Town prohibits voe (men) from entering Gerudo Town, Kara Kara Bazaar acts as a trading post and settlement where voe can stay and interact with Gerudo for the purpose of trade and/or romance between Hylians and Gerudo, as well as a place where Gerudo can spend time with their lovers/boyfriends, husbands, or fathers within Gerudo Desert as they are not permitted to live or enter Gerudo Town due to the law as only vai (women) are allowed to enter or live in Gerudo Town. However Link can sneak in by wearing Gerudo Vai clothing to disguise himself as a vai; a trick he ends up learning from Vilia. Unlike Vilia who can pass for a Gerudo due to his dark skin and height, Link is only able to pass for a Hylian Vai due to his light skin and height, thanks in part to his androgynous physique. After obtaining the Gerudo Vai outfit from Vilia, Link can purchase its piece again from Vilia should he sell it or if he wishes to have another that he can dye. In addition to Gerudo and other vai in Gerudo Town, Kara Kara Bazaar, and certain areas, Link's vai disguise also works on Hylian voe merchants and travelers he encounters outside of Gerudo Town or in Kara Kara Bazaar. Link must wear his vai disguise in order to enter Gerudo Town, even after he frees Vah Naboris.
The Inn is located inside the stone building and run by Kachoo. It provides beds and soft beds to travelers. Link can also find Rumor Mill volume 7 which contains an article about the Royal Family of Hyrule and rumors about secrets of the Royal Family that can be found in Hyrule Castle.
As Kara Kara Bazaar is a trading post, various Gerudo merchants have set up shop at Kara Kara Bazaar to sell wares to voe merchants and travelers. Additionally Beedle appears in Kara Kara Bazaar to sell his wares as there are no Stables within the Gerudo province beyond the Gerudo Canyon Stable which is located near the Gerudo Desert Gateway.
Beedle sells Arrows and ingredients that can be used to Spicy or Enduring Elixirs here. He tends to walk around during the day but doesn't go far due to the desert heat which is made worse due to the heavy backpack he has to carry. He can be found sitting outside the inn at night. Beedle will offer to trade random food dishes or elixirs for one of Link's Rhino Beetles. Link can only trade a single Rhino Beetle to Beedle once a day (in-game time) and is unable to trade if his cooking inventory is full. Link also has the option to refuse to trade.
Emri runs a stall that sells fresh fruits.
Roasted Food & Monster Part Vendor
Maike sells roasted food and a monster part. Her sister Estan is a butcher who sells roasted meats in Gerudo Town a fact Maike advertises to her customers to promote her sister's business to desert travelers (though only vai and men capable of passing for female by wearing Gerudo vai clothing like Villa and Link are able to shop there).
Role in Gerudo Culture
Due to Gerudo law prohibiting voe from entering Gerudo Town, Kara Kara Bazaar was created as a place where voe travelers and merchants can stay in the desert and where Gerudo can mingle with Hylian voe such as boyfriends, husbands, and fathers who would not be allowed in Gerudo Town, thus the Bazaar plays an important role in Gerudo culture and trade, as only vai (female) merchants and travelers are allowed (though Link manages to sneak into Gerudo Town by disguising himself as Vai). Additionally, the Bazaar would likely serve as a meeting place for Gerudo Chieftains and leaders to meet with male members of the Royal Family of Hyrule as well as male leaders of other races, as the Princesses and Queens of Hyrule would be allowed in Gerudo Town, while Kings and Princes would presumably meet with them at Kara Kara Bazaar or the leaders of the Gerudo come to Hyrule Castle like Ganondorf (presumably the Gerudo law forbidding voe does not apply to Gerudo voe due to their rarity and Gerudo tradition which implies any Gerudo voe born is destined to become King) did in Ocarina of Time, as it is highly unlikely that they would disguise themselves as Link does, especially if they were making an official diplomatic visit.