As revealed in "The End of Time", the Time Lords had become even more dangerous to the rest of reality than the Daleks were. The Daleks sought to eradicate all life in the universe, except themselves, but the Time Lords planned to eradicate even the universe. This was what Rassilon described as "the Ultimate Sanction", "the end of Time itself". According to Rassilon, the Time Lords would survive because they'd "ascend to become creatures of consciousness alone. Free of these bodies, free of time, and cause and effect, while creation itself ceases to be."
The Doctor resolved to stop both the Time Lords & the Daleks but there are 2 contradictory accounts of how he did so:
According to "The Forgotten" (comic story), the Eighth Doctor stole the Great Key of Rassilon & used it to build the Moment, which was a modified version of the De-mat Gun. With the Moment, he destroyed the Daleks & the Time Lords (together with Gallifrey), in order to protect the universe from both sides. "Don't Step on the Grass" (comic story) adds that this was what time locked the Last Great Time War.
According to "The Day of the Doctor", the War Doctor stole the Moment (also known as "the Galaxy Eater"), the last unused "forbidden" Time Lord weapon remaining in the Omega Arsenal, with the intention of using it to destroy both the Daleks and the Time Lords (as well as the whole planet of Gallifrey), in order to save the universe from both races.
The Moment, however, was so powerful & sophisticated that it had itself developed a conscience & it sought to persuade the Doctor to find another way to save the universe. "The Day of the Doctor" shows how the Moment helped the Doctor to do this. The eventual solution involved all of his first thirteen incarnations co-operating to remove Gallifrey to a "pocket universe" of its own.
Because the Dalek fleet was surrounding Gallifrey & firing on it heavily & continuously when the planet vanished, the Daleks blasted themselves almost completely out of existence (a 3-D version of the "circular firing squad"). This made it seem as if both Gallifrey & the Daleks had been destroyed. Although the Eleventh Doctor remembered that he'd saved Gallifrey, as well as the universe, the crossing of his own timeline meant that none of his earlier incarnations remembered it, so the Ninth & Tenth Doctors &, during the earlier part of his life, the Eleventh Doctor believed that he had destroyed the Time Lords along with the Daleks.
Because the Doctor was the only surviving witness to the ending of the war & because both sides had disappeared, the destruction of both sides was accepted as fact by everyone else (or, at least, by everyone else who'd been aware of the war).