The "Maesters conspiracy" theory emerged a long time ago on Reddit. As the Redditor jdylopa wrote, the theory states that the Maesters of the Citadel have been working since the beginning to rid the world from any kind of magic. They want the reason to rule the Known World and especially Westeros, therefore giving them more sway in the events to come.
What is magic?
The "GOT" world is full of magic.
We have dragons (who are believed to be tied to magic and the seasons of the world), Red Priest and Priestess, Faceless Men, Children of the Forest, White Walkers and maegi like Maggy the Frog. Think about it: a world without magic is a place where the Maesters from the Citadel are the greatest authority on all kind of matters.
There is no doubt that the Maesters hate magic. No one except them, for example, can enter the forbidden section of the library, where books about magic can be found. You may also remember how much Pycelle abhorred Qyburn's dark studies, but the best evidence for this theory can be found in George R.R.
Martin's books.
The Maesters against the Targaryen
In "A Feast for Crows" Samwell talks to Archmaester Marwyn, who is a specialist in the study of magic. Marwyn - who is ostracized by his colleagues - spent years in the east searching for lost books and studying with warlocks. This interesting man thinks there’s a conspiracy amongst the Maesters against the Targaryens. In the books, Samwell says that Maester Aemon would have probably gone to counsel Daenerys if he had the strength.
"Did he?" Marwyn replies. "Perhaps it's good that he died before he got to Oldtown. Elsewise, the [archmaesters] might have had to kill him". Why would the Maester do such thing? Because Aemon was a Targaryen. And Targaryens means dragons.
"Who do you think killed all the dragons the last time around? Gallant dragon slayers armed with swords?" the Maester tells Sam, adding that the Citadel has no place in it for sorcery or dragons.
There is more: shortly after, Sam is advised to say nothing of prophecy or other kinds of magic, since such subjects are clearly not welcome at the Citadel. According to Marwyn, those subjects could lead someone poisoning Samwell's porridge. Although we don't know many other details, if Marwyn is to be believed, the Maesters had something to do with the killing of the dragons in the last decades.
The Maesters manipulated Rickard Stark
In "A Dance with Dragons" Lady Dustin says that Rickard Stark's maester manipulated Ned's father under Maester Walys' suggestion (who was the bastard son of an Archmaester).
Walys, according to the elderly woman, wanted Rickard's heir Brandon to marry Catelyn Tully to forge an alliance that could be important in a war against Aerys II Targaryen. (Later, both Rickard and his son Brandon were burned alive by Aerys II).
What do we think about this theory
The evidence supporting the idea of a Maesters conspiracy against the Targaryen are thin. On the other hand, the conspiracy against all kind of magic looks much more plausible. Whether the Maesters's attempts to destroy magic are based on keeping the realm safe or strengthening their own power is hard to tell. Nonetheless, we think it's pretty clear that they're against any kind of sorcery. In "Game of Thrones" season 7 it's easy to see how uncomfortable with magic Archmaester Ebrose is, and we also know that the Citadel stripped Qyburn of his chain for conducting illegal experimentation.
In our opinion, the Maesters conspiracy against Targaryens may not be true, but their "conspiracy against magic" is a real thing.
Don't forget: "Game of Thrones" season 7 episode 3, titled "The Queen's Justice", airs on July 30, on HBO.