I am curious as to how is an Anbu chosen, what are the requirements, and does every one of them have like a certain backstory that makes them special or different from the normal ninjas (like how Yamato is able to do Wood style jutsu's and how Kakashi was one of the youngest Jounin in the village), or are they just trusted by the Kage's?
Anbu are recruited directly by the Hokage (or, in case of Root, by Danzou). There's no mention of a hard requirement, but generally excellent shinobi are chosen.
In general, they are chosen from the excellent jounins the village has. This is because the ANBU has to do missions like assassinating missing ninjas, and missing ninjas are more often than not, includes ninjas of jounin level. So if they are just so-so level jounin, they won't be able to do their missions well.
Of course, even though s/he is strong, doesn't necessarily qualifies them as ANBU. Notice that although Might Guy is a strong ninja, he never joined ANBU. I think one of the qualifications for entering ANBU is that even if you excels in one ability, your other abilities must be high-leveled too. Guy is very strong in taijutsu (even Uchiha Madara admitted that he never met someone remotely as good as Guy in taijutsu), but his ninjutsu and genjutsu is so so. Compared to Kakashi, although Kakashi is not as strong as Guy in taijutsu, he is quite high-leveled in it. His genjutsu is good and he is excellent in ninjutsu, being called the Copy Ninja.
The last requirement and the most important is the willingness. That they know what missions the ANBU conducts and they accept that if they join, they have to do it, even if that means you have to massacre your entire clan.
Guy actually requested to be in ANBU to keep an eye on Kakashi but the third hokage and Danzo didn't let him because they said he "Lacked the necessary darkness in his heart"