Praise be to Allah.
You do not have to take out your braces in order to fast, because nothing goes from it to the stomach. The fact that it causes you to produce more saliva does not mean that it breaks the fast.
The scholars stated that it is permissible for a person who is fasting to put a dirham (a coin) in his mouth whilst fasting, and it is more appropriate that it be permissible to put braces in one's mouth than a coin, because people do not put braces in their mouth unless they need to.
Imam Ahmad (may Allaah have mercy on him) said: Whoever puts a dirham or a dinar in his mouth whilst he is fasting, so long as the taste of it does not reach his throat, there is nothing wrong with that, but if he notices a taste, then I do not approve of that.
And if he swallows the saliva, even if it is a lot, that does not affect his fast. And Allaah knows best.