It's no secret that Sheldon (Jim Parsons) can be very difficult to deal with, so why do his friends in The Big Bang Theory tolerate him? CBS' long-running sitcom may technically be an ensemble project focusing on all seven main characters, but the series' true lead was always the socially-inept genius. Most of the primary storylines in the show was always about him and in the rare occasions that they weren't, he still played a pivotal part in it. Sheldon's quirks and idiosyncrasies were typically utilized for laughs, but for the people who actually spent time with him, these can be very frustrating.
The Big Bang Theory's original pilot significantly changed, but creators Chuck Lorre and Bill Prady were always set on anchoring the show on Sheldon and his relationship with other people, mainly his friends. Not much was changed with his characterization in the revamped show; he remained ignorant of other people's feelings and obsessed with his own intellect. Over the years, he eventually developed some social skills, but the progress was very slow. What makes Sheldon's behavior worse was that he treated his friends as poorly as he did other people. This is despite them sticking to him through his roughest patches, not to mention continue being associated with him given his weirdness.
While Howard (Simon Helberg), Bernadette (Melissa Rauch), and Raj (Kunal Nayyar) weren't as understanding, Leonard (Johnny Galecki), Penny (Kaley Cuoco), and eventually Amy (Mayim Bialik) had been patient with him — granted that there were times that they, too, were seriously irked by his behavior. Considering the number of times Sheldon had inconvenienced, demeaned, and hurt his friends, it's curious why they still choose to tolerate him. But the answer always lies in Sheldon's intentions.
Bernadette summed it up best in The Big Bang Theory season 5, episode 21 titled "The Hawking Excitation" — in the outing, Howard took advantage of Sheldon's eagerness to meet Stephen Hawking by demanding him to do humiliating tasks. This didn't sit well with Bernadette; when Howard tried to justify his actions, pointing out that Sheldon can also be mean, she tells him: "Sheldon doesn't know when he's being mean because the part of his brain that should know is getting a wedgie from the rest of his brain."
In hindsight, this makes sense and something that Leonard had operated on for years. Even when Sheldon hassled him at almost every turn and he barely got any gratitude for all the favors he'd done for him, Leonard knew that his friend wasn't doing it deliberately to make his life difficult; unfortunately, he's just wired like that. Penny also understood this as she used to point out how Sheldon would be mostly clueless when dealing with other people's emotions. In season 8, episode 18 titled "The Intimacy Acceleration," Sheldon opened up about wanting to read people's minds to her because he misinterprets others' feelings and that usually puts him in a tricky situation — this proves that he knows about his personality problem and he's hoping to solve it.
This idea was once again tackled in The Big Bang Theory season finale titled "The Stockholm Syndrome" where Amy chastised Sheldon for being self-centered amidst the varying crisis other members of the crew were going through. She told him the harsh truth that his friends only tolerate him because they're good friends and they understand that he's not aware of what he's doing — if not for those, they would've left him long ago. This resulted in Sheldon reflecting on his behavior, and eventually making up for it when he made a special shoutout to his friends during his Nobel Prize speech. Whether or not his demeanor towards his crew changed after that is unknown, but it's clear that despite his bad tendencies, he has a solid group who accepts him for who he is.