When the dragons realized that the thirteen dragons of the Forsworn had betrayed them and that they were trying to exterminate the entire dragon race, the dragons became enraged. The entire dragon race combined their powers and in one of their inexplicable feats of magic, stripped the Forsworn's dragons of their names and their ability to describe themselves. This included all names, birth names, true names, titles, thus they could not even call themselves dragons.
This had a devastating effect on the dragons of the Forsworn: they could no longer define themselves and they slowly lost the ability to think, becoming ignorant, savage beasts. The spell was so terrible that at least five of the thirteen dragons and several of the Forsworn, went mad as a result.
The dragons could also not express their personal opinions to a certain extent: that they were not able to say they disliked or liked something and could not describe themselves. It is said in Brisingr that because of these limits, they became no more than animals to ride. This generated the belief that dragons are not intelligent. It was also said that it drove many of the Forsworn mad because of it.
For example, because of the Banishing, the dragons could not say, "I like deer," or "I am red," for that would be describing, or rather, giving a name for themselves.
The only exceptions were Galbatorix's first dragon, Jarnunvösk, who was killed by Urgals, and Shruikan, whose original Rider was murdered. After the death of his original Rider, Shruikan was forcibly bonded to Galbatorix. Because Jarnunvösk was his dragon from before he became a traitor and Shruikan was forced to serve Galbatorix instead of serving him willingly, they were spared in the Banishing of Names.